Early roots
For nearly three thousand years, until the early 1800s, "algebra" primarily referred to solving polynomial equations of degree four or lower, along with the exploration of notation, the nature of their roots, and the governing laws of various number systems This era is known as classical algebra, a term that emerged in the ninth century AD By the early twentieth century, algebra transitioned into the study of axiomatic systems, leading to what is now referred to as modern or abstract algebra This significant shift from classical to modern algebra took place during the nineteenth century.
Ancient civilizations such as the Babylonians, Egyptians, Chinese, and Hindus made significant contributions to the field of mathematics by solving polynomial equations, particularly linear and quadratic equations Notably, the Babylonians, around 1700 BC, demonstrated exceptional skill in algebra, earning them recognition as proficient algebraists.
Mathematicians successfully solved quadratic equations and related equations, such as x + y = a and x² + y² = b, using methods akin to modern techniques These equations were often presented as "word problems." A typical example illustrates this approach and its corresponding solution.
I have added the area and two-thirds of the side of my square and it is 0;35 [35/60 in sexagesimal notation] What is the side of my square?
In modern notation the problem is to solve the equationx 2 +(2/3)x 5/60 The solution given by the Babylonians is:
To calculate the side of a square, start with the coefficient of 1 Calculate two-thirds of 1, which equals 0.40 Then, take half of this value, resulting in 0.20, and multiply it by itself to get 0.04 Next, add this result to 0.35, yielding 0.39 The square root of 0.50 is approximately 0.70 Finally, subtract the squared value of 0.20 from 0.70, which gives you 0.30 as the length of the square's side.
The instructions for finding the solution can be expressed in modern nota- tion as x [(0;40)/2] 2 +0;35 − (0;40)/2 = √
These instructions amount to the use of the formulax (a/2) 2 +b−a/2 to solve the equationx 2 +ax=b This is a remarkable feat See [1], [8].
The following points about Babylonian algebra are important to note:
In the absence of algebraic notation, all mathematical problems and solutions were expressed verbally This approach resulted in equations characterized by numerical coefficients, as the concept of a general quadratic equation, represented as ax² + bx + c = 0 with a, b, and c as arbitrary parameters, did not exist.
The prescriptive nature of the solutions provided clear instructions on how to achieve the desired outcomes, yet lacked justification for the procedures used However, the consistent accumulation of similar problem examples suggests that there was an underlying rationale behind the Babylonian mathematical methods employed.
The selected problems were designed to produce only positive rational number solutions, with each quadratic equation providing a single root It is believed that the Babylonian number system did not include zero, negative numbers, or irrational numbers.
The problems presented were articulated in geometric terms, yet they did not pertain to geometry itself and lacked practical application Instead, they seemed designed for student training An illustration of this is the calculation of the area in relation to two-thirds of a square's side For further insights into Babylonian algebra, refer to sources [2], [6], [14], and [18].
Around 200 BC, the Chinese and approximately 600 BC, the Indians made significant advancements in mathematics, surpassing the Babylonians They introduced the concept of negative coefficients in equations and recognized two roots for quadratic equations Although they lacked formal notation and justification for their solutions, they developed procedures for manipulating equations The Chinese, in particular, had methods for approximating roots of polynomial equations of any degree and solved systems of linear equations using early forms of "matrices," well ahead of similar techniques in Western Europe.
The Greeks
The ancient Greeks made significant advancements in mathematics, particularly in geometry and number theory, while their algebra was less developed Euclid's influential work, Elements (circa 300 BC), includes sections often viewed by historians as algebraic, despite some exceptions These sections present geometric propositions that can be translated into algebraic terms, leading to algebraic results such as laws of algebra and solutions to quadratic equations, a concept referred to as geometric algebra.
Proposition II.4 in Euclid's Elements asserts that when a straight line is divided into two parts, the square of the entire line is equal to the sum of the squares of the individual parts plus twice the area of the rectangle formed by those parts Algebraically, this can be expressed as (a+b)² = a² + 2ab + b², where 'a' and 'b' represent the segments of the line.
Proposition II.11 asserts that when a straight line is divided, the rectangle formed by the entire line and one segment will be equal to the square of the remaining segment.
The equation a(a−x)=x² requires a solution in algebraic terms In Greek algebra, the focus is on quantities rather than numbers Additionally, homogeneity in algebraic expressions is essential; all terms must share the same degree For instance, the expression x² + x = b² would not be considered a valid equation.
A much more significant Greek algebraic work is Diophantus’ Arithmetica
Around 250 AD, a significant work on number theory emerged, which also provided solutions for equations involving integers and rational numbers This text was pivotal for the development of algebra as it introduced a form of partial algebraic notation Key symbols included ς for an unknown, í for equality, σ for the square of the unknown, K for its cube, and M to indicate the absence of the unknown (similar to writing x = 0 today) For instance, the equation x³ - 2x² + 10x - 1 = 5 would be represented as K σ α ς í σ β M α í σ M ε, with letters denoting numbers—α representing 1 and ε representing 5 Notably, there was no symbol for addition, so terms with positive coefficients were listed before those with negative coefficients.
Diophantus made other remarkable advances in algebra, namely:
To effectively work with algebraic expressions, it is essential to follow two fundamental rules: first, transferring a term from one side of an equation to the other, and second, eliminating like terms from both sides of the equation.
(b) He defined negative powers of an unknown and enunciated the law of exponents, x m x n =x m + n , for−6≤m, n, m+n≤6.
(c) He stated several rules for operating with negative coefficients, for example:
“deficiency multiplied by deficiency yields availability”((−a)(−b)=ab).
He eliminated key elements of the classical Greek tradition, including the geometric interpretation of algebraic expressions, the limitation of term products to a maximum degree of three, and the necessity for homogeneity among the terms in an algebraic expression.
Al-Khwarizmi
Between the ninth and fifteenth centuries AD, Islamic mathematicians made significant advancements in algebra, with Muhammad ibn-Musa al-Khwarizmi (c 780–850) being a key figure Often referred to as “the Euclid of algebra,” al-Khwarizmi systematized the subject and established it as an independent field in his influential work, al-jabr w al-muqabalah The term “al-jabr,” which gives us the word “algebra,” involves moving negative terms in equations to the opposite side to make them positive, while “al-muqabalah” pertains to canceling equal positive terms on both sides These foundational techniques are essential for solving polynomial equations, and al-Khwarizmi applied them to quadratic equations, classifying them into five distinct types: ax² = bx, ax² = b, ax² + bx = c, ax² + c = bx, and ax² = bx + c.
The history of classical algebra reveals that al-Khwarizmi's work required a categorization of equations, as he did not recognize negative coefficients or zero His approach lacked formal notation, leading him to express problems and solutions in rhetorical terms, such as describing equations as “squares equal roots” and “squares and roots equal numbers,” with unknowns referred to as “roots.” Despite this, al-Khwarizmi provided geometric justification for his solution methods.
The following is an example of one of his problems with its solution [7, p 245]:
“What must be the square, which when increased by ten of its roots amounts to thirty- nine?” (i.e., solvex 2 +10x 9).
To solve the equation, halve the number of roots, resulting in five Multiply this by itself to get twenty-five, then add thirty-nine for a total of sixty-four Taking the square root gives you eight; subtract half the number of roots, five, to find the remainder of three This final value represents the sought-after square root Symbolically, this can be expressed as: [(1/2)×10]² + 39 − (1/2)×10.
Here is al-Khwarizmi’s justification: Construct the gnomon as in Fig 1, and
“complete” it to the square in Fig 2 by the addition of the square of side 5 The resulting square has lengthx+5 But it also has length 8, sincex 2 +10x+5 2 9+25d. Hencex=3.
In the fifteenth and sixteenth centuries, Western European mathematicians, referred to as "abacists" (derived from "abacus") and "cossists" (from the Latin word "cosa," meaning "thing" and used for unknowns), made significant contributions to the field of mathematics Their work expanded mathematical concepts and techniques, influencing the development of algebra and numerical methods during this period.
Luca Pacioli's influential work, the "Summa" of 1494, marked a significant advancement in the field of mathematics, being one of the first printed mathematics books following the invention of the printing press around 1445 His contributions included the refinement of existing notations and operational rules, which greatly improved the understanding and application of mathematical concepts.
“co” (cosa) for the unknown, introducing symbols for the first 29 (!) of its powers, “p” (piu) for plus and “m” (meno) for minus Others usedR x (radix) for square root and
R x.3 for cube root In 1557 Robert Recorde introduced the symbol “=” for equality with the justification that “noe 2 thynges can be moare equalle.” See [7], [13], [17].
Cubic and quartic equations
Around 1600 BC, the Babylonians were adept at solving quadratic equations, employing a method similar to the quadratic formula This raises the intriguing question of whether cubic equations could also be resolved with comparable formulas However, it would take another three thousand years before a definitive answer emerged, marking a significant milestone in the history of mathematics.
6 1 History of Classical Algebra in algebra when mathematicians of the sixteenth century succeeded in solving “by radicals” not only cubic but also quartic equations.
A solution by radicals for a polynomial equation provides a formula that expresses the equation's roots in relation to its coefficients This method allows for the use of basic algebraic operations—addition, subtraction, multiplication, and division—along with the extraction of roots, such as square roots and cube roots.
“radicals”) For example, the quadratic formulax = (−b±√ b 2 −4ac)/2a is a solution by radicals of the equationax 2 +bx+c=0.
The solution of cubic equations using radicals was first introduced by Cardano in his 1545 work, The Great Art of Algebra However, the method was originally discovered by del Ferro and Tartaglia, who shared their findings with Cardano under the promise of confidentiality Despite this agreement, Cardano published the method, which is now known as Cardano’s formula for solving the cubic equation \(x^3 = ax + b\), expressed as \(x = \sqrt[3]{\frac{b}{2}} + \ldots\).
Several comments are in order:
(i) Cardano used no symbols, so his “formula” was given rhetorically (and took up close to half a page) Moreover, the equations he solved all hadnumerical coefficients.
He typically found satisfaction in identifying just one root of a cubic equation By carefully selecting the appropriate cube roots, it is possible to derive all three roots of the cubic using his formula.
Negative numbers appear infrequently in his work, as he regarded them with skepticism, labeling them as "fictitious." He focused on positive coefficients and roots in cubic equations, although he accepted irrational numbers Consequently, he treated equations like x³ = ax + b and x³ + ax = b as separate entities, dedicating a chapter to the solution of each, similar to al-Khwarizmi's classification of quadratic equations.
(iv) He gavegeometricjustifications of his solution procedures for the cubic.
The resolution of fourth-degree polynomial equations, known as quartics, was achieved by reducing them to cubic equations This significant advancement was pioneered by Ferrari, whose findings were later incorporated into Cardano's work.
Methods for approximating roots of cubic and quartic equations existed prior to the radical solutions of these equations, which, despite being exact, offered limited practical utility Nonetheless, the implications of these seemingly impractical concepts from Italian Renaissance mathematicians were profoundly significant and will be explored in Chapter 2.
The cubic and complex numbers
For centuries, mathematicians believed that square roots of negative numbers could not exist, as the squares of both positive and negative numbers are positive This perspective shifted in the sixteenth century with the introduction of Cardano's formula for solving cubic equations The use of this formula naturally leads to the emergence of square roots of negative numbers, as demonstrated by applying it to the equation x³ = 9x + 2, which results in x = 3.
Cardano's skepticism towards negative numbers led him to dismiss the applicability of his formula to certain equations, such as x³ = 9x + 2 This perspective was not unfounded, as evidenced by the Pythagoreans, who considered the side of a square with an area of 2 to be nonexistent, indicating that the equation x² = 2 was deemed unsolvable in their time.
All this was changed by Bombelli In his important bookAlgebra(1572) he applied Cardano’s formula to the equationx 3 x+4 and obtainedx= 3
−121 But he could not dismiss the solution, for he noted by inspection thatx = 4 is a root of this equation Moreover, its other two roots,−2±√
The cubic equation \(x^3 + 15x + 4\) has three real roots, yet the method to derive these roots involves taking square roots of negative numbers, which was considered meaningless at the time This creates a paradox that prompts the question of how to reconcile the existence of real roots with the use of imaginary numbers in the solution process.
Bombelli adopted the rules for real quantities to manipulate “meaningless” expressions of the form a +√
−b (b > 0) and thus managed to show that 3
−1)= 4 Bombelli had given meaning to the “meaningless” by thinking the “unthinkable,” namely that square roots of negative numbers could be manipulated in a meaningful way to yield
8 1 History of Classical Algebra significant results This was a very bold move on his part As he put it:
Initially, I considered the idea to be far-fetched, sharing the common belief that it relied more on sophistry than on genuine truth However, after extensive exploration and investigation, I ultimately discovered the validity of this perspective.
Bombelli developed a “calculus” for complex numbers, stating such rules as
The concept of complex numbers emerged from the equation \(x^3 = 15x + 4\), an irreducible cubic with rational coefficients that has only real roots For two centuries, complex numbers were largely misunderstood and overlooked until their geometric representation by Gauss in 1831 established their legitimacy within the number system Earlier contributions by Argand and Wessel remained obscure among mathematicians Notably, it was demonstrated in the nineteenth century that any solution by radicals for such irreducible cubics must involve complex numbers, highlighting their essential role in solving cubic equations, contrary to the common misconception that they were primarily connected to quadratic equations, which had long been accepted as unsolvable in certain cases, such as \(x^2 + 1 = 0\).
Algebraic notation: Viète and Descartes
Mathematical notation is an essential aspect of mathematics, allowing for clear communication of complex ideas Despite its importance today, the development of symbolic notation took nearly three thousand years, with minimal symbols used initially The significant advancements in algebraic notation primarily took place during the sixteenth and early seventeenth centuries, largely attributed to the contributions of mathematicians Viète and Descartes.
The decisive step was taken by Viète in hisIntroduction to the Analytic Art(1591).
He aimed to revitalize the Greek method of analysis, a discovery approach for problem-solving, distinguishing it from their synthesis method, which is focused on proving theorems He associated this analytical method with algebra, recognizing its significance in the process of discovery.
“the science of correct discovery in mathematics,” and had the grand vision that it would “leave no problem unsolved.”
Viète revolutionized algebra by introducing arbitrary parameters into equations, distinguishing them from variables He represented parameters with consonants (B, C, D, etc.) and variables with vowels (A, E, I, etc.), allowing a quadratic equation to be expressed as BA² + CA + D = 0 This innovative approach marked a significant shift in algebra, enabling the discussion of general quadratic equations with arbitrary literal coefficients instead of relying solely on specific numerical coefficients.
Viète's groundbreaking work revolutionized algebra by shifting its focus from specific numerical equations to general polynomial equations with literal coefficients He systematically explored the relationships between the roots and coefficients of polynomial equations up to degree five, paving the way for a more abstract and symbolic approach to mathematics This transformation extended beyond algebra, establishing a symbolic science that facilitated the discovery and demonstration of mathematical results Viète's contributions were essential to the significant advancements in the seventeenth century, influencing fields such as analytic geometry, calculus, and the broader realm of mathematized science.
His work was not, however, the last word in the formulation of a fully symbolic algebra The following were some of its drawbacks:
(i) His notation was “syncopated” (i.e., only partly symbolic) For example, an equa- tion such as x 3 +3B 2 x = 2C 3 would be expressed by Viète as Acubus+
Viète emphasized the importance of "homogeneity" in algebraic expressions, requiring that all terms be of the same degree, which is why quadratic equations are often presented in an unconventional manner with all terms at the third degree This principle traces back to ancient Greek mathematics, where geometry was paramount In Greek thought, expressions like ab represented the area of a rectangle with sides a and b, while abc indicated the volume of a cube Consequently, an expression like ab + c had no meaning, as it was impossible to add lengths to areas These foundational ideas shaped mathematical practice for nearly two millennia.
The Greek legacy significantly influenced the geometric justification of algebraic results, evident in the works of mathematicians such as al-Khwarizmi and Cardano, with Viète also adhering to this tradition.
Viète limited the roots of equations to positive real numbers, a decision influenced by his geometric perspective, as there were no geometric representations available for negative or complex numbers during his time.
Most of these shortcomings were overcome by Descartes in his important book
In his seminal work "Geometry" (1637), Descartes introduced the foundational elements of analytic geometry, employing a fully symbolic notation that resembles modern mathematical conventions He designated variables with letters such as x, y, and z, while using a, b, and c for parameters A pivotal innovation was his creation of an "algebra of segments," allowing for the manipulation of line segments with operations like addition, subtraction, multiplication, and division This breakthrough eliminated the necessity for homogeneity in algebraic expressions, making combinations like ab + c valid as line segments Consequently, this marked a significant shift in mathematics, transitioning the primary language from geometry to algebra, which had dominated for two millennia.
The theory of equations and the Fundamental Theorem of Algebra 10
In the late sixteenth and early seventeenth centuries, Viète and Descartes transformed the study of equations by emphasizing theoretical approaches to polynomial equations with literal coefficients, rather than merely focusing on the solvability of numerical equations This shift led to the development of a new theory of polynomial equations, which centered on understanding the existence, nature, and quantity of their roots.
Every polynomial equation, whether it has real or complex coefficients, is guaranteed to have at least one complex root, as stated by the Fundamental Theorem of Algebra (FTA) This theorem addresses the crucial question of the existence of roots in polynomial equations, confirming that they always possess at least one solution in the complex number system.
A polynomial equation of degree n has exactly n roots, as established by Descartes' Factor Theorem This theorem states that if α is a root of the polynomial p(x), then x - α is a factor, leading to the expression p(x) = (x - α)q(x), where q(x) is a polynomial of degree one less than p(x) By applying this process iteratively, it is confirmed that a polynomial of degree n can be expressed as p(x) = (x - α₁)(x - α₂) (x - αn), where the roots α₁, α₂, , αn may not be distinct and are guaranteed to be complex numbers It is important to note that the root α of p(x) and the solution to the polynomial equation p(x) = 0 both refer to the condition p(α) = 0.
Every polynomial of odd degree with real coefficients guarantees at least one real root, a concept that was intuitively accepted in the 17th and 18th centuries and formally proven in the 19th century This principle stems from the Intermediate Value Theorem in calculus, which states that if a continuous function \( f(x) \) takes on both positive and negative values, then there exists at least one point \( x_0 \) where \( f(x_0) = 0 \).
Newton demonstrated that the complex roots of a polynomial occur in conjugate pairs, meaning if a + bi is a root, then a - bi is also a root Descartes developed an algorithm to identify all rational roots of a polynomial with integer coefficients Given a polynomial p(x) = a₀ + a₁x + + aₙxⁿ, if a/b is a rational root, where a and b are relatively prime, then a must divide a₀ and b must divide aₙ Since both a₀ and aₙ have a finite number of divisors, this method allows for the determination of all rational roots in a limited number of steps, although not every a/b that meets these conditions will necessarily be a rational root of p(x).
Descartes' Rule of Signs states that the number of positive roots of a polynomial \( p(x) \) is limited by the number of sign changes among its coefficients, while the number of negative roots cannot exceed the occurrences of two consecutive positive or negative signs.
The relationship between the roots and coefficients of a polynomial reveals that for a quadratic polynomial p(x) = ax² + bx + c, the sum of the roots (α₁ + α₂) is equal to -b/a, and the product of the roots (α₁α₂) is equal to c/a This principle, known as Viète's formulas, was later extended to polynomials of degree up to five Furthermore, Newton generalized this concept for polynomials of any degree, introducing the significant idea of symmetric functions of the roots, which express various sums and products of the roots in terms of the polynomial's coefficients.
To find the roots of a polynomial, the ideal approach is to derive an exact formula, ideally using radicals, which has been established for polynomials up to degree four Efforts to extend these formulas to higher-degree polynomials have been made In cases where exact formulas are unavailable, several methods have been developed to approximate roots with high accuracy Notable among these are Newton's method, which utilizes calculus, and Horner's method, both originating from the late seventeenth and early nineteenth centuries, respectively.
There are several equivalent versions of the Fundamental Theorem of Algebra, including the following:
(i) Every polynomial with complex coefficients has a complex root.
(ii) Every polynomial with real coefficients has a complex root.
(iii) Every polynomial with real coefficients can be written as a product of linear polynomials with complex coefficients.
(iv) Every polynomial with real coefficients can be written as a product of linear and quadratic polynomials with real coefficients.
In the early seventeenth century, Girard and Descartes provided statements of the Fundamental Theorem of Algebra (FTA), although their formulations lacked precision Descartes expressed the theorem by stating that "Every equation can have as many distinct roots as the number of dimensions of the unknown quantity in the equation," reflecting his hesitation regarding the use of complex numbers.
The Fundamental Theorem of Algebra (FTA) played a crucial role in the late seventeenth century by allowing mathematicians to determine the integrals of rational functions through the decomposition of their denominators into linear and quadratic factors While many mathematicians accepted this theorem as valid, notable figures like Gottfried Leibniz expressed skepticism In a 1702 paper, Leibniz argued that expressions such as x^4 + a^4 could not be decomposed into linear and quadratic factors, questioning the theorem's credibility.
In 1746, d'Alembert provided the first proof of the Fundamental Theorem of Algebra (FTA), which was soon followed by Euler's proof While d'Alembert's approach incorporated analytical concepts, Euler's proof was primarily algebraic Both proofs, however, were incomplete and lacked rigor, particularly in their assumption that every polynomial of degree n had roots that could be manipulated according to the rules of real numbers, ultimately asserting that the roots were complex numbers.
In his doctoral dissertation completed at just twenty years old, Gauss provided a rigorous proof of the Fundamental Theorem of Algebra (FTA) in 1797, which was published in 1799 Although his proof met the standards of his time, modern analysis reveals some gaps, as it relied on concepts from geometry and analysis Gauss later presented three additional proofs, with the second and third being primarily algebraic, culminating in his final proof in 1849.
Numerous proofs of the Fundamental Theorem of Algebra (FTA) have emerged since the 2000s, encompassing algebraic, analytic, and topological approaches This diversity is logical, as a polynomial with complex coefficients embodies characteristics of all three mathematical disciplines Interestingly, a purely algebraic proof of the FTA remains elusive, as the analytic assertion that "a polynomial of odd degree over the reals has a real root" is essential in all algebraic proofs.
In the early nineteenth century, the Fundamental Theorem of Algebra (FTA) emerged as a novel mathematical concept, demonstrating the theoretical existence of polynomial roots without providing a constructive method for finding them This nonconstructive approach sparked considerable controversy among mathematicians during the nineteenth and early twentieth centuries, with some still rejecting such existence results today For a deeper understanding, refer to sources [1], [3], [4], [5], [10], [15], and [17].
Symbolical algebra
The exploration of polynomial equations naturally involves examining the characteristics and properties of different number systems, as the solutions to these equations are numerical in nature Consequently, understanding number systems is a crucial component of classical algebra.
In the eighteenth century, negative and complex numbers became essential due to the Fundamental Theorem of Algebra, yet they were often met with skepticism and a lack of understanding Notably, Isaac Newton characterized negative numbers as quantities that prompted confusion among mathematicians of his time.
Leibniz described complex numbers as “an amphibian between being and nonbeing,” while Euler categorized quantities as positive when marked with a plus sign and negative when marked with a minus sign.
Historically, the manipulation of negative numbers, such as the rule (−1)(−1)=1, lacked justification despite being recognized since ancient times Mathematicians like Euler suggested that (−a)(−b) must equal ab, as it could not be −ab In the late 18th and early 19th centuries, members of the Analytical Society at Cambridge University sought to understand the rationale behind these rules At Cambridge, mathematics was integrated into liberal arts education and regarded as a model of absolute truths essential for logical reasoning Consequently, these mathematicians aimed to establish a solid foundation for algebra, particularly regarding the operations involving negative numbers.
Peacock's 1830 work, "Treatise of Algebra," is a seminal contribution to the field, introducing the distinction between "arithmetical algebra" and "symbolical algebra." Arithmetical algebra involves operations on symbols representing only positive numbers, which Peacock argued required no justification For instance, the expression a−(b−c)=a−b+c exemplifies a law of arithmetical algebra when b > c and a > b−c.
It becomes a law of symbolical algebra if no restrictions are placed ona, b, andc.
Symbolical algebra, a newly established field, focuses on operations with symbols that do not necessarily represent specific objects but adhere to the principles of arithmetic algebra This approach allowed Peacock to formally define various algebraic laws, such as demonstrating that the product of two negative numbers, (−a)(−b), is equal to the positive product ab.
Since(a−b)(c−d)=ac+bd−ad−bc(**) is a law of arithmetical algebra when- evera > bandc > d, it becomes a law of symbolical algebra, which holds without restriction ona, b, c, d Lettinga=0 andc=0 in (**) yields(−b)(−d)=bd.
Peacock sought to align the laws of symbolical algebra with those of arithmetical algebra through the Principle of Permanence of Equivalent Forms, asserting that the principles of symbolical algebra should mirror those of arithmetical algebra Although the specific laws were not clearly defined at the time, they were later established as axioms for rings and fields in the late nineteenth century This foundational idea parallels modern axiomatic approaches to algebra, highlighting a significant conceptual shift in the discipline's focus.
14 1 History of Classical Algebra on the meaning of symbols to a stress on their laws of operation Witness Peacock’s description of symbolical algebra:
In symbolic algebra, the operations are governed by established rules that define their meaning These rules can be viewed as arbitrary assumptions, as they are imposed on a system of symbols and their combinations, which could be tailored to fit any other coherent set of rules.
Peacock's innovative concept was advanced for its era, yet he merely acknowledged the arbitrary nature of laws without fully embracing it Ultimately, the principles of arithmetic continued to dominate Over the following decades, English mathematicians began to implement Peacock's ideas by developing algebras that exhibited distinct properties from traditional arithmetic As noted by Bourbaki, this shift marked a significant evolution in mathematical thought.
Between 1830 and 1850, English algebraists introduced the abstract concept of composition laws, significantly expanding the scope of Algebra They applied this concept to various mathematical objects, including Boole's algebra of Logic, Hamilton's vectors and quaternions, and Cayley's matrices and non-associative laws.
Despite its limitations, symbolic algebra fostered a conducive environment for future advancements in the field The symbols and operational laws evolved into subjects of independent study, transcending their original role as mere representations of numerical relationships This evolution will be explored in the following chapters.
1 I G Bashmakova and G S Smirnova,The Beginnings and Evolution of Algebra, The
MathematicalAssociation ofAmerica, 2000 (Translated from the Russian byA Shenitzer.)
2 I G Bashmakova and G S Smirnova, Geometry: The first universal language of math- ematics, in: E Grosholz and H Breger (eds),The Growth of Mathematical Knowledge,
3 N Bourbaki,Elements of the History of Mathematics, Springer-Verlag, 1991.
4 D E Dobbs and R Hanks,A Modern Course on the Theory of Equations, Polygonal
5 B Fine and G Rosenberg,The Fundamental Theorem of Algebra, Springer-Verlag, 1987.
6 J Hoyrup, Lengths, Widths, Surfaces: A Portrait of Babylonian Algebra and its Kin,
7 V Katz,A History of Mathematics, 2nd ed., Addison-Wesley, 1998.
8 V Katz, Algebra and its teaching: An historical survey,Journal of Mathematical Behavior
9 I Kleiner, Thinking the unthinkable: The story of complex numbers (with a moral),
10 M Kline,Mathematical Thought from Ancient to Modern Times, Oxford University Press,
11 P G Nahin,An Imaginary Tale: The Story of√
12 K H Parshall, The art of algebra from al-Khwarizmi to Viète: A study in the natural selection of ideas,History of Science1988,26: 129–164.
13 H M Pycior, George Peacock and the British origins of symbolical algebra,Historia Mathematica1981,8: 23–45.
14 E Robson, Influence, ignorance, or indifference? Rethinking the relationship between Babylonian and Greek mathematics,Bulletin of the British Society for the History of MathematicsSpring 2005,4: 1–17.
15 H W Turnbull,Theory of Equations, Oliver and Boyd, 1957.
16 S Unguru, On the need to rewrite the history of Greek mathematics,Archive for the History of Exact Sciences1975–76,15: 67–114.
17 B L van der Waerden,A History of Algebra, from al-Khwarizmi to Emmy Noether,
18 B L van der Waerden,Geometry and Algebra in Ancient Civilizations, Springer-Verlag,
19 B L van der Waerden, Defence of a “shocking” point of view,Archive for the History ofExact Sciences1975–76,15: 199–210.
This chapter explores the foundational concepts and theories of group theory, including abstract groups, normal subgroups, quotient groups, simple groups, free groups, isomorphisms, homomorphisms, automorphisms, composition series, and direct products Key theorems such as Lagrange's, Cauchy's, Cayley's, and Jordan-Hölder are also discussed, alongside the theories of permutation groups and abelian groups.
Group theory evolved within the broader context of mathematics, particularly algebra, with its development beginning in 1770 and reaching significant milestones throughout the nineteenth century This era was characterized by several key mathematical trends, including a heightened emphasis on rigor, the rise of abstraction, the revival of the axiomatic method, and the perception of mathematics as a human endeavor independent of physical contexts.
Until the late eighteenth century, algebra primarily focused on solving polynomial equations However, by the twentieth century, it evolved into the study of abstract, axiomatic systems This shift from classical algebra to modern algebra took place in the nineteenth century, giving rise to various structures such as group theory, commutative rings, fields, noncommutative rings, and vector spaces These concepts often developed together, as seen in Galois theory, which integrates groups and fields, and algebraic number theory, which combines group theory with commutative ring and field theory Additionally, group representation theory encompasses elements of group theory, noncommutative algebra, and linear algebra.
Sources of group theory
Classical Algebra
In 1770, Lagrange addressed significant issues in algebra in his memoir “Reflections on the Solution of Algebraic Equations,” focusing on polynomial equations He explored theoretical questions regarding the existence and nature of roots, such as whether every equation has a root and the characteristics of these roots, including their quantity and types—real, complex, positive, or negative Additionally, he examined practical questions related to methods for finding these roots, distinguishing between exact and approximate methods, with a particular emphasis on the former.
The Babylonians effectively solved quadratic equations using the method of completing the square as early as 1600 BC By 1540, algebraic techniques for solving cubic and quartic equations were established However, the algebraic resolution of quintic equations remained a significant challenge for two centuries, a problem that Lagrange aimed to address in his 1770 paper.
In this paper, Lagrange examines the methods developed by Viète, Descartes, Euler, and Bezout for solving cubic and quartic equations He highlights that these methods share a common approach: reducing the original equations to auxiliary equations, known as resolvent equations, which are one degree lower than the original.
Lagrange extended his analysis to polynomial equations of arbitrary degree by associating a resolvent equation with each original equation For a given polynomial \( f(x) \) with roots \( x_1, x_2, \ldots, x_n \), he introduced a rational function \( R(x_1, x_2, \ldots, x_n) \) based on the roots and coefficients of \( f(x) \) By examining the distinct values that \( R \) takes under all \( n! \) permutations of the roots, he defined the resolvent equation as \( g(x) = (x - y_1)(x - y_2) \cdots (x - y_k) \), where \( y_1, y_2, \ldots, y_k \) are the resulting values.
The coefficients of the polynomial g(x) are symmetric functions of the variables x₁, x₂, x₃, , xₙ, indicating that they can be expressed as polynomials in the elementary symmetric functions of these variables This relationship highlights that the coefficients are derived from the original equation f(x) Lagrange demonstrated that k divides n!, which is the foundation of what is now known as Lagrange’s theorem in group theory.
In the case of a quartic polynomial with roots \( x_1, x_2, x_3, x_4 \), the resolvent function \( R(x_1, x_2, x_3, x_4) \) can be defined as \( x_1 x_2 + x_3 x_4 \), which yields three distinct values across the twenty-four permutations of the roots Consequently, the resolvent equation for a quartic is cubic in nature However, when extending this analysis to quintic polynomials, Lagrange discovered that the resolvent equation is of degree six.
Lagrange's efforts, while not solving the quintic's algebraic solvability, marked a significant milestone in mathematics by linking polynomial equation solutions to the permutations of their roots This association became a cornerstone of his general theory of algebraic equations, highlighting the importance of root permutations in understanding polynomial behavior.
Lagrange's exploration of the solution of equations laid the groundwork for the concept of groups, particularly through his discussion of permutations, despite lacking a formal calculus of permutations His insights were later vindicated by Galois, highlighting the foundational role of permutation groups in the development of modern algebra For further details, refer to sources [12], [16], [19], [25], and [33].
Number Theory
In his 1801 work, "Disquisitiones Arithmeticae," Gauss synthesized and unified prior developments in number theory, while also proposing new avenues of research that captivated mathematicians throughout the century.
The Disquisitiones initiated the theory of finite abelian groups, with Gauss establishing key properties of these groups without employing modern group theory terminology These groups manifested in four distinct forms: the additive group of integers modulo m, the multiplicative group of integers relatively prime to m modulo m, the group of equivalence classes of binary quadratic forms, and the group of n-th roots of unity.
And although these examples turned up in number-theoretic contexts, it is as abelian groups that Gauss treated them, using what are clear prototypes of modern algebraic proofs.
For example, considering the nonzero integers modulop(pa prime), he showed that they are all powers of a single element; that is, that the groupZ ∗ of such integers
19 is cyclic Moreover, he determined the number of generators of this group, showing that it is equal toϕ(p−1), whereϕis Euler’sϕ-function.
In his exploration of the group structure of \( Z_p^* \), he defined the order of an element and demonstrated that this order is a divisor of \( p-1 \) Utilizing this finding, he proved Fermat's Little Theorem, which states that \( a^{p-1} \equiv 1 \mod p \) for any integer \( a \) not divisible by \( p \), thereby applying group-theoretic concepts to derive number-theoretic conclusions Furthermore, he established that for any positive integer \( t \) that divides \( p-1 \), there exists an element in \( Z_p^* \) with an order of \( t \), effectively presenting a converse to Lagrange's theorem in the context of cyclic groups.
In his exploration of the then-th roots of 1 linked to the cyclotomic equation, he demonstrated that these roots also constitute a cyclic group He posed and addressed numerous questions regarding this group, paralleling the inquiries he made concerning the group Z ∗ p.
The representation of integers by binary quadratic forms dates back to Fermat in the early 17th century, notably illustrated by his theorem that every prime of the form 4n+1 can be expressed as the sum of two squares, x² + y² Gauss further expanded on this topic in his work "Disquisitiones," dedicating significant effort to the comprehensive study of binary quadratic forms and their ability to represent integers.
A binary quadratic form is represented by the expression ax² + bxy + cy², where a, b, and c are integers Gauss introduced a composition method for these forms, denoting the composition of two forms K₁ and K₂ as K₁ + K₂ He demonstrated that this composition is both associative and commutative, established the existence of an identity element, and confirmed that each form has an inverse These findings confirm that the set of binary quadratic forms forms an abelian group, showcasing their mathematical structure and properties.
Despite these remarkable insights, one should not infer that Gauss had the concept of an abstract group, or even of a finite abelian group Although the arguments in the
Disquisitionesare quite general, each of the various types of “groups” he considered was dealt with separately—there was no unifying group-theoretic method which he applied to all cases.
Geometry
We are referring here to Klein’s famous and influential (but see [18]) lecture entitled
“A Comparative Review of Recent Researches in Geometry,” which he delivered in
Felix Klein introduced the Erlangen Program in 1872, upon his admission to the University of Erlangen's faculty, with the primary aim of classifying geometry as the study of invariants under various transformation groups This groundbreaking program involved the examination of groups such as the projective, rigid motions, similarities, hyperbolic, and elliptic groups, as well as their associated geometries Notably, the affine group was not included in Klein's original Erlangen Program, which laid the foundation for a new approach to understanding geometry.
The nineteenth century witnessed an explosive growth in geometry, both in scope and in depth New geometries emerged: projective geometry, noneuclidean geometries, differential geometry, algebraic geometry,n-dimensional geometry, and
Grassmann’s geometry of extension Various geometric methods competed for supremacy: the synthetic versus the analytic, the metric versus the projective.
In the mid-20th century, a significant challenge emerged regarding the classification of relationships and connections among various geometries and geometric methods This led to the exploration of "geometric relations," emphasizing properties of figures that remain unchanged under transformations Over time, the emphasis transitioned from geometric relations to the transformations themselves, transforming the study of geometric figures into an examination of their associated transformations.
Different types of transformations, such as collineations, circular transformations, inversive transformations, and affinities, have been the focus of specialized studies This exploration prompted investigations into the logical relationships between these transformations, ultimately leading to the classification of transformations and culminating in Klein's group-theoretic synthesis of geometry.
Klein's incorporation of groups in geometry marked a significant advancement in the organization of the field This development followed the establishment of the Cayley–Sylvester Invariant Theory in the 1850s, which aimed to analyze the invariants of forms under variable transformations This classification theory served as a precursor to Klein's Erlangen Program and can be considered implicitly group-theoretic In contrast, Klein's application of groups in geometry was explicit, highlighting a crucial evolution in geometric understanding For a detailed examination of the implicit group-theoretic concepts leading to Klein's work, refer to [33].
The significance of Klein’s Erlangen Program and his other contributions is crucial for understanding the evolution of group theory Originating a century after Lagrange's work and eighty years after Gauss's, the importance of the Program can be fully appreciated by examining the historical development of group theory, starting from Lagrange and Gauss and culminating around 1870.
Analysis
In 1874, Lie introduced his general theory of continuous transformation groups, which are now known as Lie groups These groups are represented by transformations of the form x_i = f_i(x_1, x_2, , x_n, a_1, a_2, , a_n), where f_i are analytic functions of the variables x_i and parameters a_i, which can be either real or complex An example of such transformations is x_1 = (ax + b)/(cx + d), where a, b, c, and d are real numbers and the condition ad - bc = 0 holds, illustrating the concept of a continuous transformation group.
Lie viewed himself as the successor to Abel and Galois, aiming to revolutionize differential equations in the same way they transformed algebraic equations His inspiration stemmed from the realization that many differential equations solvable by traditional methods exhibit invariance under specific continuous groups This insight prompted him to explore differential equations that maintain invariance under a designated continuous group and to examine the simplifications these equations might undergo due to the known characteristics of the group, akin to concepts in Galois theory.
“Galois theory of differential equations,” his work was fundamental in the subsequent formulation of such a theory by Picard (1883-1887) and Vessiot (1892).
In the late 19th century, specifically around 1876, mathematicians Poincaré and Klein initiated their research on automorphic functions, which are extensions of various elementary functions such as circular, hyperbolic, and elliptic functions These functions are defined as analytic functions of a complex variable z within a specific domain D, exhibiting invariance under a particular group of transformations represented by the equation x₁ = (ax + b) / (cx + d), where a, b, c, and d can be real or complex numbers and must satisfy the condition ad - bc = 0 Additionally, the transformations must form a "discontinuous" group, meaning that within any compact domain, only a finite number of transformations can be applied to any given point.
The modular group, consisting of integers a, b, c, and d satisfying ad - bc = 1, is linked to elliptic modular functions, while Fuchsian groups, with real numbers a, b, c, and d under the same condition, relate to Fuchsian automorphic functions Following Klein's Erlangen Program, we will examine the implications of these findings for group theory in the subsequent section.
Development of “specialized” theories of groups
Permutation Groups
Lagrange's 1770 work marked the beginning of the study of permutations in relation to solving equations, representing a significant early instance of implicit group-theoretic thinking in mathematics This foundational work paved the way for subsequent contributions from Ruffini, Abel, and Galois in the early nineteenth century, ultimately leading to the development of the concept of a permutation group.
Ruffini and Abel demonstrated the unsolvability of the quintic equation by expanding on Lagrange's concepts regarding resolvents, which indicated that a necessary condition for solving a general polynomial equation of degree n is the existence of a resolvent of degree less than n They concluded that such resolvents do not exist for n greater than 4, contributing to the development of permutation theory However, it was Galois who made significant conceptual advancements in this field and is widely regarded as the founder of permutation group theory.
Familiar with the contributions of Lagrange, Abel, and Gauss on polynomial equations, he sought more than just a solvability method; he aimed to uncover fundamental principles Expressing his dissatisfaction with the increasingly complex computational methods of his time, he noted, “From the beginning of this century, computational procedures have become so complicated that any progress by those means has become impossible.”
2.2 Development of “specialized” theories of groups
Galois distinguished between "Galois theory," which explores the relationship between fields and groups, and its practical applications in solving equations He emphasized that he was presenting "the general principles and just one application" of the theory However, many early commentators overlooked this distinction, resulting in a focus on applications that overshadowed the theoretical foundations of Galois theory.
Évariste Galois was the pioneer in using the term "group" in a technical context, defining it as a collection of permutations that are closed under multiplication He noted that if substitutions S and T are in the same group, the substitution ST must also be present Galois identified that key properties of algebraic equations correspond to specific characteristics of a group linked to the equation, which he termed "the group of the equation." To articulate these properties, he introduced the essential concept of normal subgroup, applying it with significant impact.
Galois shifted the focus from resolvent equations, which were complex and lacked a clear methodology, to the group associated with the equation and its subgroups He observed that the existence of a resolvent was equivalent to having a normal subgroup of prime index within the equation's group, highlighting a more systematic approach to understanding the problem.
Galois defined the group of an equation as follows:
Given an equation with roots a, b, c, etc., there exists a set of permutations of these roots that possesses two key properties: first, every function of the roots that remains unchanged under these permutations is a rational function of the coefficients and any additional quantities; second, any function of the roots that can be expressed as a rational function is invariant under these permutations.
The group of an equation is defined as the set of permutations of its roots that maintain all relations among the roots within the field of coefficients While this definition does not ensure the existence of such a group, Galois proved its existence and explored how the group evolves when new elements are added to the ground field His approach closely aligns with contemporary algebraic methods.
Galois’ work was slow in being understood and assimilated In fact, while it was done around 1830, it was published posthumously by Liouville, in 1846 Beyond his technical accomplishments,
Galois significantly influenced future developments in mathematics by presenting theorems that lacked clear proofs and defined concepts, prompting his successors to address these gaps Additionally, it was essential not only to validate these theorems but also to extract their fundamental group-theoretic essence.
For details see [12], [19], [23], [25], [29], [31], [33] See also Chapter 8.3.
In the early nineteenth century, Cauchy emerged as a significant contributor to permutation theory, establishing permutation groups as an independent field of study through his influential papers in 1815 and 1844 Prior to Cauchy's work, permutations were primarily seen as tools for solving polynomial equations rather than subjects of focused research While he recognized the contributions of Lagrange and Ruffini, Cauchy's approach was not directly influenced by the contemporary group-theoretic perspectives on the solvability of algebraic equations, as Galois' work had yet to be published.
Cauchy's 1815 papers marked the first systematic exploration of permutation groups, where he acknowledged the significance of sets of permutations closed under multiplication, although he did not assign a specific name to them He introduced the term “diviseur indicatif” to refer to the number of elements within such a closed set, highlighting its importance in the study of permutation groups.
In the 1844 paper he defined the concept of a group of permutations generated by certain elements:
In the context of substitutions involving elements such as x, y, and z, the products resulting from these substitutions are referred to as derived substitutions Together with the original substitutions, these derived substitutions create a framework known as a system of conjugate substitutions.
2.2 Development of “specialized” theories of groups
Cauchy's influential papers made significant contributions to permutation theory, introducing essential terminology and notation still in use today, such as permutation and cyclic notation He defined key concepts including the product of permutations, degree of a permutation, cyclic permutation, and transposition, while recognizing the identity permutation as a valid permutation Additionally, Cauchy explored what is now known as the direct product of two groups and extensively examined alternating groups, showcasing a variety of important results in the field.
(i) Every even permutation is a product of 3-cycles.
(ii) Ifp(prime) is a divisor of the order of a group, there exists a subgroup of orderp.
This is known today asCauchy’s theorem, though it was stated without proof by
(iii) Determines all subgroups ofS 3 , S 4 , S 5 , andS 6 (making an error inS 6 ).
(iv) All permutations which commute with a given one form a group, nowadays called thecentralizerof an element of the group.
It should be noted that all these results were given and proved in the context of permutation groups For details see [6], [8], [23], [24], [25], [33].
Jordan's 1870 Treatise on Substitutions and Algebraic Equations stands as a pivotal contribution to the fields of Galois theory and group theory While the preface claims the work aims to enhance Galois' method and demonstrate its effectiveness in addressing key problems in equation theory, the core focus of the treatise is actually on group theory itself, rather than merely its application to solvability of equations.
Jordan's work exemplifies the quest for a mathematical synthesis rooted in key concepts, a pursuit shared by contemporaries like Klein He identified the notion of a permutation group as a pivotal idea, allowing him to unify results from notable mathematicians such as Galois and Cauchy By applying the group concept across various fields, including the theory of equations, algebraic geometry, transcendental functions, and theoretical mechanics, Jordan contributed to a cohesive understanding of mathematics His exploration encompassed all areas of algebraic geometry, number theory, and function theory, focusing on the intriguing applications of permutation groups Ultimately, his work serves as a comprehensive review of contemporary mathematics, highlighting the significance of group-theoretic thinking in permutation-theoretic contexts.
Abelian Groups
The foundation of abelian group theory can be traced back to number theory, particularly through Gauss' "Disquisitiones Arithmeticae," with implicit contributions from Euler’s number-theoretic work Unlike permutation theory, the group-theoretic concepts in number theory were not explicitly recognized until the late 19th century, as the term "group" was not utilized, and its connection to the developing theory of permutation groups was absent This article highlights examples of implicit group-theoretic principles in number theory, focusing on algebraic number theory.
Algebraic number theory emerged from the study of Fermat’s Last Theorem and Gauss’ theory of binary quadratic forms, focusing on algebraic number fields and their arithmetic properties In 1846, Dirichlet investigated the units in these fields, proving that the group of units can be expressed as a direct product of a finite cyclic group and a free abelian group of finite rank Concurrently, Kummer introduced “ideal numbers” and established an equivalence relation on them, leading to the discovery of special properties related to the class number of cyclotomic fields, which corresponds to the order of the ideal class group Dirichlet had previously conducted similar research on quadratic fields.
In 1869, Schering, a former student of Gauss, explored the structure of Gauss' equivalence classes of binary quadratic forms He identified essential fundamental classes that allow for the derivation of all other forms through composition In group-theoretic language, Schering established a basis for the abelian group representing the equivalence classes of binary quadratic forms.
2.2 Development of “specialized” theories of groups
In his 1870 paper titled “An exposition of some properties of the class number of ideal complex numbers,” Kronecker expanded upon Kummer’s research on cyclotomic fields, applying it to arbitrary algebraic number fields He approached the subject from a highly abstract perspective, examining a finite set of arbitrary elements and establishing an abstract operation that adhered to specific laws, which can now be recognized as the axioms of a finite abelian group.
Let θ₁, θ₁₁, and θ₁₁₁ be a finite set of elements where any two can be combined to produce a third element through a defined procedure f For any elements θ₁ and θ₁₁, the result of the procedure f(θ₁, θ₁₁) yields a unique θ₁₁₁ This procedure exhibits properties such as commutativity (f(θ₁, θ₁₁) = f(θ₁₁, θ₁)) and associativity (f(θ₁, f(θ₁₁, θ₁₁₁)) = f(f(θ₁, θ₁₁), θ₁₁₁)) Moreover, if θ₁₁ is distinct from θ₁₁₁, then f(θ₁, θ₁₁) will also be unique By adopting multiplication notation (θ₁ * θ₁₁) in place of f(θ₁, θ₁₁) and using equivalence instead of strict equality, we can express the relationship θ₁ * θ₁₁ ∼ θ₁₁₁ through the equation f(θ₁, θ₁₁) = θ₁₁₁.
Kronecker focused on establishing the laws governing the combination of "magnitudes," which led to an implicit definition of a finite abelian group From these abstract principles, he derived several significant consequences.
(i) Ifθis any “element” of the set under discussion, thenθ k =1 for some positive integerk Ifkis the smallest such, thenθis said to “belong tok.” Ifθbelongs to kandθ m =1, thenkdividesm.
(ii) If an elementθbelongs tok, then every divisor ofkhas an element belonging to it.
(iii) Ifθ andθ 1 belong tok andk 1 respectively, andkandk 1 are relatively prime, thenθ θ 1 belongs tokk 1
A fundamental system of elements, denoted as θ 1, θ 2, θ 3, and so on, allows for the unique representation of each element in a given set through the expression θ 1 h 1 θ 2 h 2 θ 3 h 3, where h i equals 1, 2, 3, and so forth, for n i The numbers n 1, n 2, n 3, etc., corresponding to each θ, are structured such that each number is divisible by its successor Moreover, the product of these numbers, n 1 n 2 n 3, collectively equals the total number of elements in the set.
The established framework by Kronecker can be interpreted in relation to finite abelian groups, particularly highlighting (iv) as the basis theorem for these groups He applied this framework to specific instances, including equivalence classes of binary quadratic forms and ideal classes Notably, when applying (iv) to binary quadratic forms, it leads to Schering’s result.
Kronecker, while not explicitly linking his implicit definition of finite abelian groups to the established concept of permutation groups, acknowledged the benefits of the abstract perspective he embraced.
The very simple principles .are applied not only in the context indicated but also frequently elsewhere—even in the elementary parts of number theory.
These principles exist within a broader and more abstract framework of ideas To facilitate their development, it is essential to remove any unnecessary constraints, allowing for a more streamlined application of the argument across various scenarios.
Also, when stated with all admissible generality, the presentation gains in simplicity and, since only the truly essential features are thrown into relief, in transparency [33].
In 1879, Frobe- nius and Stickelberger published a significant paper titled “On groups of commuting elements,” which built upon Kronecker’s work while explicitly incorporating the concept of abelian groups They advanced the understanding of abstract group theory by recognizing its connections to congruences, Gauss’ composition of forms, and Galois substitution groups Additionally, they acknowledged groups of infinite order, such as the units of number fields and all roots of unity A key outcome of their research was the proof of the basis theorem for finite abelian groups, including the uniqueness of decomposition, which contrasts with Kronecker's earlier formulation.
A non-irreducible group can be broken down into purely irreducible factors, and this decomposition can occur in various ways Despite the different methods of decomposition, the total number of irreducible factors remains constant Furthermore, it is possible to pair the factors from different decompositions such that corresponding factors share the same order.
The researchers identified "irreducible factors" as cyclic groups with prime power orders They subsequently applied their findings to various mathematical structures, including groups of integers modulo m, binary quadratic forms, and ideal classes within algebraic number fields.
Frobenius and Stickelberger's paper presents a significant advancement in the independent theory of finite abelian groups, establishing its foundations in a manner that aligns with contemporary perspectives For further details, refer to sources [5], [9], [24], [30], and [33].
Transformation Groups
In the late nineteenth century, group-theoretic concepts began to emerge explicitly in geometry and analysis, largely influenced by Klein and Lie This marked a significant shift in group theory, moving the focus from permutation groups to groups of transformations While permutation groups continued to be studied, this transition highlighted a growing interest in infinite groups over finite ones.
Klein acknowledged the relationship between his work and permutation groups while recognizing the divergence he was pursuing He emphasized that both Galois theory and his own research focus on the exploration of "groups of changes," highlighting a shared foundational concept despite their differences.
2.2 Development of “specialized” theories of groups
Felix Klein (1849–1925) highlighted the distinction between Galois theory, which focuses on a finite set of discrete elements, and his own work, which addresses an infinite number of elements within a continuous manifold He emphasized the importance of developing a theory of transformations, paralleling the existing theory of permutation groups, to explore groups generated by specific types of transformations.
Klein shunned the abstract point of view in group theory, and even his technical definition of a (transformation) group is deficient:
A sequence of transformations, denoted as A, B, C, and so on, is classified as a group of transformations if the composite of any two transformations within the sequence results in another transformation that also belongs to the same sequence.
Klein significantly expanded the understanding of groups and their relevance across various mathematical disciplines, advocating for the idea that group-theoretic concepts are essential to the foundation of mathematics.
The special subject of group theory extends through all of modern mathe- matics As an ordering and classifying principle, it intervenes in the most varied domains.
There was another context in which groups were associated with geometry, namely
Motion geometry involves the application of motions or transformations of geometric objects as group elements In 1856, Hamilton implicitly examined "groups" of regular solids, while Jordan focused on classifying all subgroups of the motion group in Euclidean 3-space in 1868 Additionally, Klein, in his 1884 Lectures on the Icosahedron, utilized the symmetry group of the icosahedron to solve the quintic equation, revealing a profound relationship between geometric groups.
29 rotations of the regular solids, polynomial equations, and complex function theory.
In theseLecturesthere also appeared the “Klein 4-group.”
In the late 1860s Klein and Lie had jointly undertaken “to investigate geometric or analytic objects that are transformed into themselves bygroups of changes.” (This is
In 1894, Klein provided a retrospective overview of his program, focusing primarily on discrete groups In contrast, Lie explored continuous transformation groups, recognizing their significant potential in geometry and differential equations Lie aimed to identify all groups of continuous transformations, highlighting the importance of this theory in advancing mathematical understanding.
[continuous] transformations.” He achieved his objective by the early 1880s with the classification of these groups A classification of discontinuous transformation groups was obtained by Poincaré and Klein a few years earlier.
The foundational significance of the extensive theories developed in discontinuous and continuous transformation groups lies not only in their technical accomplishments but also in their ongoing relevance as active fields of research.
(i) They provided a major extension of the scope of the concept of a group—from permutation groups and abelian groups to transformation groups;
(ii) They introduced important examples of infinite groups—previously the only objects of study were finite groups;
The group concept has been significantly expanded to encompass a wide array of applications, including number theory, algebraic equations, geometry, and both ordinary and partial differential equations, as well as function theory, which covers automorphic and complex functions.
Before the concept of an abstract group was established, significant developments laid the groundwork for its emergence In the following sections, we will delve deeper into the definition and implications of abstract groups For additional information, please refer to sources [5], [7], [9], [17], [18], [20], [24], [29], and [33].