A brief history of simple groups
The study of non-abelian finite simple groups dates back to Galois around 1830, who recognized their crucial role as obstacles in solving polynomial equations using radicals Galois understood the importance of classifying these groups and identified that the alternating groups \(A_n\) are simple for \(n \geq 5\) Additionally, he constructed simple groups such as \(PSL_2(p)\) for prime numbers \(p \geq 5\).
Every finite groupGhas acomposition series
The Jordan–Hölder theorem asserts that in a composition series of groups, where each group is normal in the next and cannot be refined further, the composition factors \( G_i / G_{i-1} \) are unique and independent of the series chosen, meaning that if one series has a non-abelian composition factor, all do Additionally, Galois's theorem indicates that a polynomial equation in one variable can be solved by radicals if and only if its corresponding Galois group possesses a composition series with cyclic factors.
The 19th century experienced gradual advancements in finite group theory, notably marked by Camille Jordan's 'Traité des substitutions' in 1870, which introduced constructions of the simple groups now recognized as PSL n (p) This was complemented by the publication of Sylow's theorems in 1872, providing essential tools for the classification of simple groups Additionally, Mathieu's remarkable 1861 paper laid the groundwork for the sporadic groups M11 and M12, followed by his 1873 work that constructed M22, M23, and M24.
It was not until the dawn of the 20th century that a well-developed theory of the finite classical groups began to emerge, most notably in the work of
R.A Wilson,The Finite Simple Groups,
Graduate Texts in Mathematics 251, ©Springer-Verlag London Limited 2009
L E Dickson In large part this work was inspired by Killing’s classification of complex simple Lie algebras (if that is not an oxymoron) into the types
A n ,B n ,C n ,D n ,G2,F4,E6,E7 andE8 Dickson constructed finite simple groups analogous to all of these except F4, E7 and E8, for every finite field (with a small number of exceptions which are not simple).
Dickson's failure to construct finite simple groups of the remaining three types raises questions, particularly since it took another fifty years for Chevalley to provide a uniform construction of these groups in his renowned 1955 paper The classical groups derived from types A_n, B_n, C_n, and D_n include PSL_{n+1}(q) (linear), PΩ_{2n+1}(q) (orthogonal, odd dimension), PSp_{2n}(q) (symplectic), and PΩ^+_{2n}(q) (orthogonal, even dimension, plus type) However, the absence of unitary groups and PΩ_{-2n}(q) was soon addressed through a ‘twisting’ of Chevalley's construction This led to the independent creation of two new families, 3D_4(q) and 2E_6(q), by Steinberg, Tits, and Hertzig, using the unitary groups as a model.
Soon afterwards, Suzuki and Ree saw how to ‘twist’ the groups of typesB2,
In the early 1960s, there was a prevailing belief that the finite simple groups G2 and F4, with field characteristics of 2, 3, and 2 respectively, had been fully discovered, leaving only the task of proving this assertion.
In the early 1960s, significant advancements in group theory emerged, particularly with the Feit–Thompson paper of 1963, which established that every finite group of odd order is soluble, implying that every non-abelian finite simple group must have even order and contain an involution This led to the audacious idea that all finite simple groups could be classified through induction, with Thompson providing the foundational case by classifying minimal simple groups However, the quest for a complete classification proved to be more challenging than anticipated, especially following Janko's 1964 construction of a simple group with an involution centralizer C2 × A5, which raised doubts about the feasibility of the classification project and suggested the possibility of countless undiscovered simple groups.
In the decade following the initial discoveries, an additional twenty sporadic simple groups, as termed by Burnside, were identified, leading to a sense of nearing completion in the classification of finite simple groups by 1980 Despite this belief, interest in discovering new groups persisted, with claims of nearing proof and premature predictions regarding the decline of group theory circulating among researchers.
The Classification Theorem
The Classification Theorem for Finite Simple Groups (CFSG) asserts that every finite simple group is isomorphic to one of a specific set of groups This theorem emphasizes the significance of understanding the structure of finite simple groups rather than merely the classification itself.
(i) a cyclic groupC p of prime orderp;
(ii) an alternating groupA n , forn5;
(iii) a classical group: linear: PSL n (q),n2, except PSL2(2) and PSL2(3); unitary: PSU n (q),n3, except PSU3(2); symplectic: PSp 2n (q),n2, except PSp 4 (2); orthogonal: PΩ2n+1(q),n3,qodd;
(iv) an exceptional group of Lie type:
(v) one of 26 sporadic simple groups:
• the seven Leech lattice groups Co1, Co2, Co3, McL, HS, Suz, J2;
• the three Fischer groups Fi22, Fi23, Fi 24 ;
• the five Monstrous groupsM,B, Th, HN, He;
• the six pariahs J1, J3, J4, O’N, Ly, Ru.
Conversely, every group in this list is simple, and the only repetitions in this list are:
This book aims to comprehensively explain the classification of finite simple groups (CFSG) by introducing all finite simple groups, providing concrete constructions when feasible, calculating their orders, proving their simplicity, and examining their actions on various natural structures.
In this article, we explore geometrical and combinatorial objects to uncover significant aspects of subgroup structures Our research demonstrates a considerable portion of the converse of the Classification of Finite Simple Groups (CFSG), specifically confirming the existence of numerous finite simple groups.
In this book, the term 'construction' is frequently employed to refer to the process of building an object related to the group in question, even if it lacks formal proof However, it is important to note that this discussion does not address the proof of the central aspect of the Classification of Finite Simple Groups (CFSG), specifically the assertion regarding the non-existence of any other finite simple groups.
Applications of the Classification Theorem
After decades of classifying objects, researchers may lose sight of their original motivations, as seen in the 1980 predictions about the decline of group theory However, group theorists dedicated significant effort to classifying simple groups not merely for display, but to deepen understanding and further the field.
The initial significant applications of CFSG (Classification of Finite Simple Groups) were found in permutation group theory, particularly in the classification of multiply-transitive groups, which led to the identification of the first five sporadic groups by Mathieu The O’Nan–Scott Theorem marked a pivotal advancement in classifying primitive permutation groups, simplifying the task to the classification of maximal subgroups of index n within almost simple groups This highlighted the necessity for a comprehensive classification of maximal subgroups in simple groups While complete lists of maximal subgroups can be derived for individual groups, such as the sporadic groups—where all except the Monster have been calculated—the same is not always feasible for families of groups In the case of alternating groups, the O’Nan–Scott theorem provides some explicit maximal subgroups and indicates that any other maximal subgroup is almost simple and acts primitively Furthermore, a theorem by Liebeck, Praeger, and Saxl clarifies the conditions under which certain subgroups are not maximal, yet an explicit list of the remaining subgroups cannot be provided.
In 1984, Aschbacher initiated a program to classify the maximal subgroups of classical groups, building on previous work, including L E Dickson's classification of PSL2(q) over a century ago Currently, explicit lists of maximal subgroups are known for small dimensions, aided by comprehensive representations of quasisimple groups in dimensions up to 250, as compiled by Hiss, Malle, and Lübck This progress raises the possibility of creating explicit lists of maximal subgroups for classical groups in these dimensions Additionally, while complete lists exist for five of the ten families of exceptional groups of Lie type, there is hope for further advancements in this area.
Remarks on the proof of the Classification Theorem
The debate surrounding whether the CFSG should be classified as a theorem has sparked broader philosophical discussions about the nature of theorems, proofs, and mathematics itself Most mathematicians tend to be pragmatic, acknowledging that achieving a perfect proof with absolute certainty is unrealistic In fact, those who advocate for the absolute nature of proofs often produce work that does not meet this ideal Ultimately, the discussion about whether an argument qualifies as a proof is less meaningful than the degree of certainty it provides Throughout the twentieth century, significant mathematical problems, including the CFSG, were claimed to be solved, but skepticism is essential as not all claims have endured Typically, a process of expert review and correction leads to the eventual acceptance of these solutions, yet the quest for absolute certainty remains elusive due to human fallibility and the potential for long-hidden mistakes.
The status of the CFSG remains a topic of discussion, as the process of compiling and refining the proof has taken significantly longer than anticipated since its initial announcement in 1980 Gorenstein, Lyons, and Solomon are currently working on documenting the complete proof, with six out of an expected eleven volumes already published.
The 'quasithin' case is addressed in two separate volumes by Aschbacher and Smith, totaling around 1200 pages, while the existence and uniqueness of the 26 sporadic simple groups remain unquestioned Most aspects of the proof have been reviewed and re-proven by various researchers, significantly reducing the risk of catastrophic errors, though it cannot be entirely ruled out.
Prerequisites
This book aims to minimize prerequisites, assuming familiarity with abstract group theory, including Sylow's theorems and the Jordan-Hölder theorem, as well as fundamental concepts in linear algebra, similar to those found in Kaye and Wilson A reasonable level of mathematical maturity is expected, and while basic knowledge of representation theory from James and Liebeck is beneficial for some proofs, it is not essential for most of the text Additionally, in the chapter on sporadic groups, I will occasionally reference basic properties of graphs, codes, lattices, and other mathematical constructs, which I hope can be understood through context For clarity, a summary of the assumed background in group theory is provided for those who may need it.
A group is a finite set \( G \) that includes an identity element \( 1 \), a binary multiplication operation \( x \cdot y \), and a unary inverse \( x^{-1} \), all of which must satisfy the associative law, identity laws, and inverse laws for all elements \( x, y, z \) in \( G \) A group is classified as abelian if the multiplication is commutative, meaning \( xy = yx \) for all \( x, y \in G \); otherwise, it is non-abelian A subgroup \( H \) is a subset of \( G \) that is closed under the operations of multiplication and taking inverses To verify that \( H \) is a subgroup, it is sufficient to check that \( xy^{-1} \in H \) for all \( x, y \in H \) The left cosets of \( H \) in \( G \) are defined as \( gH = \{ gh \mid h \in H \} \), while the right cosets are \( Hg = \{ hg \mid h \in H \} \) Both left and right cosets have the same size and partition the group \( G \), leading to the relationship \( |G| = |H| \cdot |G:H| \).
Lagrange’s Theorem states that the order of a finite group G, denoted as |G|, is related to the index of a subgroup H in G, represented as |G : H|, which indicates the number of left or right cosets Additionally, the order of an element g within G is defined as the order n of the cyclic group generated by g, denoted as C_n, consisting of the elements {1, g, g^2, , g^(n-1)} Understanding these concepts is essential for exploring the structure and properties of groups in abstract algebra.
G is the lowest common multiple of the orders of the elements, that is the smallest positive integer esuch thatg e = 1 for allg∈G.
A homomorphism is a mapping φ: G → H that preserves multiplication, satisfying the condition φ(xy) = φ(x)φ(y), which implies φ(1) = 1 and φ(x^(-1)) = φ(x)^(-1) The kernel of φ, denoted as kerφ, consists of elements g in G for which φ(g) = 1, forming a subgroup that maintains the equality of its left and right cosets, thus qualifying as a normal subgroup An isomorphism is a bijective homomorphism where kerφ = {1} and φ(G) = H, indicating that G and H are structurally identical, represented as G ≅ H.
In group theory, if N is a normal subgroup of G, the quotient group G/N consists of elements of the form xN for each x in G, with group operations defined as (xN)(yN) = (xy)N and (xN)⁻¹ = x⁻¹N According to the first isomorphism theorem, for a homomorphism φ: G → H, the image of φ is isomorphic to the quotient group G/ker(φ), with the isomorphism established by the mapping φ(x) to x(ker(φ)).
Normal subgroups of G/N correspond one-to-one with normal subgroups K of G that contain N, as stated in the second isomorphism theorem, which asserts that (G/N)/(K/N) is isomorphic to G/K For any subgroup H of G and normal subgroup N of G, the set HN = {xy | x ∈ H, y ∈ N} forms a subgroup of G, while the intersection N ∩ H is a normal subgroup of H This relationship is further elaborated in the third isomorphism theorem.
Simple groups and composition series
A group S is simple if it has exactly two normal subgroups (1 and S) In particular, an abelian group is simple if and only if it has prime order A series
1 =G 0 G 1 G 2 ã ã ãG n − 1 G n =G (1.3) for a group G is called a composition series if all the factors G i /G i − 1 are simple (and they are then called composition factors).
Thefourth isomorphism theorem(orZassenhaus’s butterfly lemma) states that if XY GandABGthen
According to the Jordan-Hölder Theorem, any two series for a finite group \( G \) have isomorphic refinements, leading to the conclusion that all composition series share the same composition factors, including their multiplicities A normal series consists of terms \( G_i \) that are normal in \( G \), and if such a series has no proper refinements, it is referred to as a chief series, with its factors \( G_i / G_{i-1} \) being identified as chief factors.
A group is considered soluble if it possesses a composition series where all composition factors are abelian, which means they are cyclic of prime order In group theory, a commutator, represented as [x, y], is defined as the element x⁻¹y⁻¹xy The subgroup generated by all such commutators from elements x and y in the group G is known as the commutator subgroup or derived subgroup, denoted as [G, G].
G WritingG (0) =GandG (n) = (G (n − 1) ) , it follows thatGis soluble if and only ifG (n) = 1 for somen AlsoG/N is abelian if and only ifN containsG , so G/G is thelargest abelian quotientofG.
Group actions and conjugacy classes
The regular representation of a group G involves associating each element g in G with the permutation x → xg of the elements in G This illustrates that every finite group is isomorphic to a group of permutations, as stated in Cayley’s theorem Additionally, if G acts as a group of permutations on a set Ω, the stabilizer of an element a in Ω is defined as the subgroup H, which includes all permutations in G that map the element a to itself.
8 1 Introduction itself Then Lagrange’s theorem can be re-interpreted as the orbit–stabiliser theorem, that |G|/|H| equals the number of images of a under G (i.e the lengthof theorbitofa).
In group theory, the action of a group \( G \) on itself via conjugation defines orbits known as conjugacy classes, denoted as \([x] = \{g^{-1} x g | g \in G\}\) The stabilizer of an element \( x \) is identified as the centralizer \( C_G(x) = \{g \in G | g^{-1} x g = x\} \) Conjugacy classes serve to partition the group \( G \), with their sizes being divisors of the order of \( G \) An element \( x \) belongs to a conjugacy class of size 1 if it commutes with every element of \( G \), indicating that \( x \) is in the center \( Z(G) = \{y \in G | g^{-1} y g = y \text{ for all } g \in G\} \), which is a normal subgroup of \( G \) This framework is particularly relevant in the study of \( p \)-groups and nilpotent groups.
A finite group is termed a p-group if its order is a power of the prime p, which implies that all elements within the group have an order that is also a power of p, according to Lagrange’s Theorem Each conjugacy class in a p-group contains p^a elements for some integer a, ensuring that there are at least p conjugacy classes of size 1, thus indicating that the center Z(G) has an order of at least p We define Z^1(G) as Z(G) and Z^n(G)/Z^(n-1)(G) as Z(G/Z^(n-1)(G)) If G is a p-group, then there exists an n such that Z^n(G) equals G A group exhibiting this characteristic is referred to as nilpotent, with a class at most n, leading to a specific series structure.
1 =Z 0 (G)Z 1 (G)Z 2 (G)ã ã ã is called the upper central series.
The direct product G₁ × × Gₖ of groups G₁, , Gₖ is defined on the set {(g₁, , gₖ) | gᵢ ∈ Gᵢ} with group operations given by (g₁, , gₖ)(h₁, , hₖ) = (g₁h₁, , gₖhₖ) and the inverse as (g₁, , gₖ)⁻¹ = (g₁⁻¹, , gₖ⁻¹) A finite group is considered nilpotent if and only if it can be expressed as a direct product of p-groups.
If m and n are coprime, then the group C_m × C_n is isomorphic to C_mn This implies that in any finite abelian group, there exists an element whose order matches the group's exponent Furthermore, every finite abelian group can be expressed as a direct product of cyclic groups, C_{n_1} × C_{n_2} × × C_{n_r}, where each integer n_i divides n_{i-1} for all 2 ≤ i ≤ r Notably, the integers n_i are uniquely determined by the structure of the group.
IfGis a finite group of orderp k n, wherepis prime andnis not divisible by p, then theSylow theoremsstate that
(ii) theseSylow p-subgroupsare all conjugate; and
(iii) the number s p of Sylow p-subgroups satisfies s p ≡ 1 modp (Note also that, by the orbit–stabiliser theorem,s p is a divisor of n).
To demonstrate the first statement, consider the action of G on all subsets of size p^k through right multiplication; since the number of these subsets is not divisible by p, there exists a stabilizer with an order divisible by p^k, thus equal to p^k For the second statement, to show that any p-subgroup Q is contained within a Sylow p-subgroup, we analyze the action of Q on the right cosets P_g of a Sylow p-subgroup P via right multiplication Given that the number of cosets is not divisible by p, we find an orbit {P_g} of length 1, leading to the conclusion that P_gQ = P_g and gQg^(-1) lies within P Lastly, to prove the third statement, we let a Sylow p-subgroup P act by conjugation on all other Sylow p-subgroups; the orbits must have lengths divisible by p; otherwise, P and Q would represent distinct Sylow p-subgroups of N_G(Q), resulting in a contradiction.
An important corollary of Sylow’s theorems is the Frattini argument: if
NGandP is a Sylowp-subgroup ofN, then G=N G (P)N.
An automorphism of a group G is defined as an isomorphism from G to itself, and the collection of all such automorphisms forms a group known as Aut(G) Within this framework, inner automorphisms are represented by the function φ_g: x → g⁻¹xg for any g in G, creating a subgroup called Inn(G) within Aut(G) Notably, Inn(G) is a normal subgroup of Aut(G) since for any α in Aut(G), the relation α⁻¹φ_gα = φ_gα holds Additionally, the composition of inner automorphisms satisfies φ_gh = φ_gφ_h, and φ_g equals φ_h if and only if g*h⁻¹ is an element of the center Z(G) of G Consequently, the mapping φ: g → φ_g establishes a homomorphism from G onto Inn(G), with the kernel being the center Z(G).
Therefore Inn(G)∼=G/Z(G) and, in particular, ifZ(G) = 1 thenG∼= Inn(G).Theouter automorphism groupofGis Out(G) = Aut(G)/Inn(G).
Notation
There is currently no universal agreement on the notation used for simple groups and their extensions In this book, I will primarily adhere to the notation established in the 'Atlas of Finite Groups,' although there are exceptions Notably, for orthogonal groups, I prefer to use Dieudonné's notation to avoid potential misunderstandings associated with the Atlas Detailed notation for simple groups can be found in Section 1.2, and extensions of groups will be presented in several specified formats.
A×B denotes a direct product, with normal subgroups Aand B; alsoA:B denotes a semidirect product (or split extension), with a normal subgroup
A and a subgroupB; andA.B denotes a non-split extension, with a normal subgroup A and quotient B, but no subgroup B; finally A.B or just AB denotes an unspecified extension.
The expression [n] denotes an (unspecified) group of order n, while nor
In group theory, C_n typically represents a cyclic group of order n, while for a prime p, p^n denotes an elementary abelian group of order p^n, which is formed as a direct product of n copies of C_p Additionally, q^n, where q is a power of p, is sometimes used to denote an elementary abelian p-group, though this usage is not part of the standard Atlas notation This definition also encompasses the scenario where n equals 1.
Introduction
L E Dickson In large part this work was inspired by Killing’s classification of complex simple Lie algebras (if that is not an oxymoron) into the types
A n ,B n ,C n ,D n ,G2,F4,E6,E7 andE8 Dickson constructed finite simple groups analogous to all of these except F4, E7 and E8, for every finite field (with a small number of exceptions which are not simple).
Dickson's failure to construct finite simple groups of the remaining three types remains a mystery, especially since it took another fifty years for Chevalley to provide a uniform construction in his influential 1955 paper The classical groups, including PSL n+1 (q), PΩ 2n+1 (q), PSp 2n (q), and PΩ + 2n (q), correspond to types A n, B n, C n, and D n respectively However, the absence of unitary groups and PΩ − 2n (q) prompted the realization that these could be derived by 'twisting' Chevalley's construction Inspired by unitary groups, Steinberg, Tits, and Hertzig independently introduced two new families: 3 D 4(q) and 2 E 6(q).
Soon afterwards, Suzuki and Ree saw how to ‘twist’ the groups of typesB2,
In the early 1960s, it was widely believed that all finite simple groups had been identified, with G2 and F4 being characterized by fields of 2, 3, and 2, respectively The prevailing sentiment at the time suggested that the only task left was to validate this assumption.
In the early 1960s, significant advancements in group theory emerged, highlighted by the 1963 Feit–Thompson paper, which established that every finite group of odd order is soluble, implying that non-abelian finite simple groups must have even order and contain an involution This breakthrough led to the ambitious endeavor of classifying all finite simple groups, with Thompson providing the foundational case by identifying minimal simple groups However, the complexity of this classification project became evident when Janko introduced a simple group with an involution centralizer \( C_2 \times A_5 \) in 1964, challenging the assumption that the classification would be straightforward and raising questions about the existence of countless undiscovered simple groups.
In the decade following the initial discoveries, an additional twenty sporadic simple groups, as termed by Burnside in reference to the Mathieu groups, were identified However, by 1980, the momentum of discoveries had significantly slowed, leading to a widespread belief that the classification of finite simple groups was nearing completion, with few, if any, remaining to be found Despite this prevailing sentiment, some researchers continued their search, leading to announcements of nearly complete proofs and premature predictions regarding the decline of group theory.
The Classification Theorem for Finite Simple Groups (CFSG) asserts that every finite simple group is isomorphic to one of a specific set of groups This theorem highlights the significance of understanding the structure of finite simple groups rather than merely focusing on the classification itself.
(i) a cyclic groupC p of prime orderp;
(ii) an alternating groupA n , forn5;
(iii) a classical group: linear: PSL n (q),n2, except PSL2(2) and PSL2(3); unitary: PSU n (q),n3, except PSU3(2); symplectic: PSp 2n (q),n2, except PSp 4 (2); orthogonal: PΩ2n+1(q),n3,qodd;
(iv) an exceptional group of Lie type:
(v) one of 26 sporadic simple groups:
• the seven Leech lattice groups Co1, Co2, Co3, McL, HS, Suz, J2;
• the three Fischer groups Fi22, Fi23, Fi 24 ;
• the five Monstrous groupsM,B, Th, HN, He;
• the six pariahs J1, J3, J4, O’N, Ly, Ru.
Conversely, every group in this list is simple, and the only repetitions in this list are:
This book aims to elucidate the statement of CFSG by introducing all finite simple groups, offering concrete constructions where feasible, calculating their orders, proving their simplicity, and examining their actions on various natural structures.
In this article, we explore geometrical and combinatorial objects to uncover significant aspects of subgroup structures Our findings demonstrate a considerable portion of the converse part of the Classification of Finite Simple Groups (CFSG), specifically confirming the existence of numerous finite simple groups.
In this book, the term 'construction' is used in a broader sense, referring to the creation of an object related to the group in question, rather than strictly as an 'existence proof.' However, it is important to note that this text does not address the proof regarding the core aspect of the Classification of Finite Simple Groups (CFSG), specifically the non-existence of any other finite simple groups.
1.3 Applications of the Classification Theorem
After decades of classifying certain objects, it's easy to lose sight of the original motivation behind the project The predictions of the decline of group theory in 1980 reflected a misunderstanding of its purpose Group theorists dedicated significant effort to classifying simple groups not merely to display them, but to deepen understanding and advance the field.
The application of CFSG in permutation group theory has led to significant advancements, particularly in classifying multiply-transitive groups, a problem that was pivotal in Mathieu's discovery of the first five sporadic groups The classification of 2-transitive groups followed, while the O’Nan–Scott Theorem marked a key development in classifying primitive permutation groups by focusing on the maximal subgroups of almost simple groups This highlighted the necessity for a comprehensive classification of maximal subgroups within simple groups Although complete lists of maximal subgroups can be generated for individual groups, such as sporadic groups—where all but the Monster have been cataloged—this is not always feasible for families of groups For instance, the O’Nan–Scott theorem provides some explicit maximal subgroups for alternating groups, yet it also indicates that any other maximal subgroup is almost simple and acts primitively, with a theorem by Liebeck, Praeger, and Saxl detailing conditions under which such subgroups are not maximal, but an exhaustive list remains elusive.
In 1984, Aschbacher initiated a program for classifying the maximal subgroups of classical groups, building on earlier work that included L E Dickson's classification of PSL2(q) over a century ago For small dimensions, comprehensive lists of maximal subgroups are established, and with the extensive representation data of quasisimple groups up to dimension 250 provided by Hiss, Malle, and Lübck, there is potential for creating explicit lists of maximal subgroups for classical groups in these dimensions Furthermore, while complete lists exist for five out of ten families of exceptional groups of Lie type, there is optimism for further advancements in this area.
1.4 Remarks on the proof of the Classification Theorem
The debate surrounding whether the Classification of Finite Simple Groups (CFSG) qualifies as a theorem has sparked discussions about the nature of theorems, proofs, and mathematics itself Most mathematicians adopt a pragmatic approach and do not aspire to achieve the unattainable ideal of a perfect proof that guarantees absolute certainty While some assert the necessity of absolute proof, many of their own contributions may not meet this standard Ultimately, it is more meaningful to discuss the degree of certainty derived from an argument rather than its classification as a proof The twentieth century witnessed the announcement of solutions to several long-standing mathematical problems, including CFSG, the four-colour problem, Fermat’s Last Theorem, and the Poincaré conjecture While skepticism is warranted as not all solutions endure scrutiny, a process of expert review often leads to general acceptance of these resolutions However, the quest for absolute certainty remains elusive, as mathematicians are inherently fallible and prone to errors that can persist undetected for years.
The status of the CFSG remains a topic of interest, as the comprehensive proof has proven to be more complex than initially anticipated since its announcement in 1980 The ongoing project by Gorenstein, Lyons, and Solomon aims to compile the entire proof into a cohesive format, with six out of the planned eleven volumes already published.
The 'quasithin' case is not included in this series but has been extensively addressed in two volumes by Aschbacher and Smith, totaling around 1200 pages Additionally, the existence and uniqueness of the 26 sporadic simple groups are not discussed, but this aspect is well-established and not in doubt Over time, various researchers have revisited and re-proven most parts of the proof, significantly reducing the chances of catastrophic errors, though they cannot be entirely ruled out.
This book aims to minimize prerequisites, assuming familiarity with abstract group theory, including Sylow’s theorems and the Jordan–Hölder theorem, as well as foundational linear algebra, similar to Kaye and Wilson's work A reasonable level of mathematical maturity is expected While most of the text does not require knowledge of representation theory, some proofs may reference concepts from James and Liebeck Additionally, the chapter on sporadic groups will occasionally utilize basic properties of graphs, codes, lattices, and other mathematical objects, which should be understandable from the context For clarity, a summary of the assumed background in group theory is provided for those interested.
Permutations
The alternating groups
An alternative interpretation of the picture involves analyzing it from the bottom to the top and noting the positions of the swapped strings In this example, we first swap the second and third strings, followed by the third and fourth, and finally the fourth and fifth As a result, the second string moves to the fifth position, the third string to the second, and so forth This approach allows us to express our permutation as a series of adjacent string swaps Furthermore, the combination of two permutations can be represented by concatenating the corresponding lists of swapped strings.
When expressing the identity permutation as a product of transpositions, if the lines connecting the indices cross, they must cross back, indicating that the number of crossings for the identity element is even Consequently, if a permutation π is represented in two different ways as a product of transpositions, both representations will either contain an even number of transpositions or an odd number This leads to the conclusion that the mapping φ from the symmetric group S n to the group {±1}, defined by φ(π) = 1 for permutations composed of an even number of transpositions, is a well-defined group homomorphism.
The kernel of the onto homomorphism φ is a normal subgroup of index 2, known as the alternating group of degree n, which consists of even permutations This group has an order of 1/2 n!, while the remaining elements of S n are classified as odd permutations For additional insights, an alternative proof demonstrating that A n has an index of 2 in S n can be referenced in Exercise 2.1.
The notation for permutations can be confusing, as it differs from the conventional function notation To clarify, we adopt a new notation where we write a π instead of π(a), ensuring that a πρ is interpreted as ρ(π(a)) This approach allows us to read permutations from left to right, aligning with standard reading practices, rather than the right-to-left interpretation used for functions.
Transitivity
In the context of a subgroup H of a symmetric group S n, we examine the mapping capabilities of H on a set Ω If every point in Ω can be mapped to every other point by elements of H, we define H as transitive on Ω This is formally expressed by the condition that for all points a and b in Ω, there exists a permutation π in H such that a π = b The collection of points that can be reached from a specific point a through the permutations in H is referred to as the orbit of H containing a.
It is easy to see that the orbits of H form a partition of the setΩ.
A group is termed k-transitive if it can map k distinct points to any other set of k distinct points Specifically, for any two lists of k distinct points, there exists an element π in the group such that each point in the first list is mapped to the corresponding point in the second list Notably, 1-transitive groups are equivalent to transitive groups.
For example, it is easy to see that the symmetric groupS n isk-transitive for allkn, and that the alternating groupA n isk-transitive for allkn−2.
If a group H is k-transitive when k equals 1, it follows that H is (k−1)-transitive and thus m-transitive for all m greater than k An intriguing concept that lies between 1-transitivity and 2-transitivity is primitivity, which warrants its own exploration and is best understood by clarifying what it does not represent.
Primitivity
A block system for a subgroup H ofS n is a partition of Ω preserved by H; that is, a set of mutually disjoint non-empty subsets ofΩ whose union isΩ.
In the context of partitions, the components are referred to as blocks If two points, a and b, belong to the same block, then for any element π in the group H, the points aπ and bπ will also be in the same block There are two fundamental block systems that every group maintains: the trivial block system, which consists of a single block Ω, and the trivial block system where each block contains only one point A non-trivial block system is known as a system of imprimitivity for the group H; if n ≥ 3, any group with such a system is termed imprimitive, while a non-trivial group without it is labeled primitive Additionally, S2 is typically classified as primitive, whereas S1 is considered neither primitive nor imprimitive.
It is obvious that ifH is primitive, thenH is transitive (2.1)
If H = 1 is not transitive, then the orbits of H create a system of imprimitivity, indicating that H is not primitive Conversely, there are many transitive groups that are not primitive, such as the subgroup in S4.
The transitive property of the group preserves the block systems {{1,2},{3,4}}, {{1,3},{2,4}}, and {{1,4},{2,3}} In any block system associated with a transitive imprimitive group, all blocks maintain uniform size.
Another important basic result about primitive groups is that every 2-transitive group is primitive (2.2)
If the group H is imprimitive, we can select three distinct points, a, b, and c, where a and b belong to the same block, while c is in a different block This scenario is feasible because each block contains at least two points, and there are a minimum of two blocks Consequently, there cannot be an element of H that maps the pair (a, b) to the pair (a, c), indicating that H is not 2-transitive.
Group actions
In the context of group theory, let G be a subgroup of the symmetric group S_n that acts transitively on the set Ω The stabilizer of a specific point a in Ω, denoted H, is defined as H = {g ∈ G | a g = a} Consequently, there exists a natural bijection between the points of Ω and the right cosets of H in G, represented as Hg.
This bijection is given byHx↔a x It is left as an exercise for the reader (see Exercise2.2) to prove that this is a bijection In particular,|G:H|=n.
We can turn this construction around, so that given any subgroupH inG, we can letGact on the right cosets ofH according to the rule (Hx) g =Hxg.
By numbering the cosets of H from 1 to n, where n represents the index |G : H|, we create a permutation action of G on these n points This establishes a group homomorphism from G to the symmetric group S_n When this homomorphism is injective, it indicates that G acts faithfully.
Maximal subgroups
The relationship between transitive group actions and subgroups allows for valuable translations between the combinatorial characteristics of the set Ω and the group-theoretical attributes of G Notably, a primitive group action is associated with a maximal subgroup, highlighting the interplay between these mathematical concepts.
H of Gis calledmaximal if there is no subgroupK withH < K < G More precisely:
Proposition 2.1 Suppose that the group G acts transitively on the set Ω, and letH be the stabiliser ofa∈Ω ThenGacts primitively onΩif and only if H is a maximal subgroup ofG.
Proof We prove both directions of this in the contrapositive form First as- sume that H is not maximal, and choose a subgroup K with H < K < G.
The points of Ω correspond to the right cosets of H in G Additionally, the cosets of K in G are unions of H-cosets, which correspond to sets of points, with each set containing |K : H| points Furthermore, the action of G preserves the set of these points.
K-cosets, so the corresponding sets of points form a system of imprimitivity forGonΩ.
If group G acts imprimitively, let Ω 1 represent the block containing element a in the imprimitivity system Given that G is transitive, the stabilizer of Ω 1 acts transitively on it, but not on the entire set Ω Consequently, this stabilizer is a proper subgroup of G that strictly contains H, indicating that H is not a maximal subgroup.
Wreath products
The concept of imprimitivity leads naturally to the idea of a wreath product of two permutation groups Recall thedirect product
G×H ={(g, h)|g∈G, h∈H} (2.3) with identity element 1 G × H = (1 G ,1 H ) and group operations
Recall also the semidirect product G:H or G: φ H, where φ : H → Aut(G) describes an action ofH onG We defineG:H={(g, h)|g∈G, h∈H} with identity element 1 G:H = (1 G ,1 H ) and group operations
In this context, let H represent a permutation group acting on the set Ω = {1, , n} We define G n as the direct product of n copies of the group G, expressed as G n = G × G × × G, where each g i belongs to G The group H acts on G n by permuting the subscripts of the n elements This action is formalized through a homomorphism φ: H → Aut(G n ), which is defined by the mapping φ(π − 1)(g 1, , g n) = (g 1 π, , g n π).
The wreath product \( GH \) is defined as \( G \nabla \phi H \) For instance, when \( H \) is isomorphic to \( S_n \) and \( G \) is isomorphic to \( S_m \), the wreath product \( S_m \nabla S_n \) can be constructed by taking \( n \) copies of \( S_m \), each acting on distinct sets \( \Omega_1, \ldots, \Omega_n \) of size \( m \) Subsequently, the subscripts \( 1, \ldots, n \) are permuted by elements of \( H \), resulting in an imprimitive action.
The sum of the sets Ω can be expressed as S m S n on Ω = n i=1 Ω i, maintaining the partition of Ω into the subsets Ω i In a broader context, any transitive imprimitive group can be represented as a wreath product When considering the blocks of a system of imprimitivity for the group G, denoted as Ω 1, , Ω k, it is evident that all subsets Ω i are of equal size, and thus G serves as a subgroup of Sym(Ω 1)S k.
Simplicity
Cycle types
A permutation π can be represented using an alternative notation that highlights its cycles Each cycle is formed by selecting an element a from the set Ω, which is then mapped by π to a π, followed by a π 2, and so forth Since Ω is finite, this process will eventually lead to a repetition, resulting in a relation a π j = a π k, indicating that a returns to its original position The first occurrence of this repetition defines the cycle, which can be expressed as (a, a π, a π 2, , a π k−1), where k denotes the cycle's length By applying this method to another element b not included in the initial cycle, we can derive an additional disjoint cycle of π This process continues until all elements of Ω are exhausted, allowing π to be represented as a product of disjoint cycles.
The cycle type of a permutation represents the lengths of its cycles, typically abbreviated for convenience For instance, the identity permutation is characterized by the cycle type (1^n), while a transposition has the cycle type (2, 1^(n-2)) It's important to note that a cycle with an even length corresponds to an odd permutation, and conversely, a permutation is classified as even if it contains an even number of cycles with even lengths.
In the symmetric group S n, if ρ is another permutation, the mapping π ρ = ρ − 1 πρ transforms a into a πρ Each cycle of π, represented as (a, a π, a π², , a π k−1), corresponds to a cycle (a ρ, a πρ, a π² ρ, , a π k−1 ρ) in π ρ, indicating that the cycle types of π and π ρ are identical Conversely, if two permutations π and σ share the same cycle type, their cycles can be paired by length, allowing for the definition of a permutation ρ that maps elements from one cycle to the corresponding elements in the other, leading to the conclusion that two permutations in S n are conjugate if and only if they possess the same cycle type.
To find the centraliser of a permutation π within the symmetric group S_n, we perform the operation of conjugating π by itself For a permutation π of cycle type (c_1 k_1, c_2 k_2, , c_r k_r), the centraliser in S_n is expressed as a direct product of r groups, specifically C_{c_i} S_{k_i}.
Conjugacy classes in the alternating groups
In analyzing the conjugacy classes within the alternating group A_n, it is essential to identify which elements are centralized by odd permutations For any element g in A_n and an odd permutation ρ, there are two scenarios: either gρ is conjugate to g by some element π in A_n, or it is not If they are conjugate, then g is centralized by the odd permutation ρπ^(-1) Conversely, if they are not conjugate, all odd permutations will map g into the same A_n-conjugacy class as gρ, indicating that no odd permutation centralizes g.
If a permutation g has a cycle of even length, it is centralized by that cycle, which is classified as an odd permutation Likewise, if g contains two cycles of the same odd length, it is centralized by an element ρ that interchanges these two cycles; since ρ is the product of an odd number of transpositions, it is also categorized as an odd permutation.
If a group \( g \) lacks an even cycle or two odd cycles of the same length, it can be expressed as a product of disjoint cycles of distinct odd lengths Each element \( \rho \) that centralizes \( g \) must map each cycle to itself, allowing the first point in each cycle to be assigned arbitrarily while the positions of the remaining points are subsequently determined Consequently, all such elements \( \rho \) can be represented as products of powers of the cycles of \( g \), confirming that \( \rho \) is an even permutation.
The centralization of g by no odd permutation occurs if and only if g is expressed as a product of disjoint cycles with distinct odd lengths Consequently, the conjugacy classes of A_n are linked to cycle types when there is at least one cycle of even length or when two cycles share the same length In contrast, a cycle type comprising distinct odd lengths corresponds to two separate conjugacy classes in A_n For instance, in A_5, the cycle types of even permutations include (1 5) and (3, 1 2).
(2 2 ,1), and (5) Of these, only (5) consists of disjoint cycles of distinct odd lengths Therefore there are just five conjugacy classes in A 5.
The alternating groups are simple
A subgroup H of a group G is considered normal if it encompasses entire conjugacy classes within G A group G is classified as simple if it contains only two normal subgroups: the trivial subgroup and itself Additionally, every non-abelian simple group G is perfect, meaning that G is equal to its own derived subgroup.
The numbers of elements in the five conjugacy classes inA 5are 1, 20, 15,
12 and 12 respectively Since no proper sub-sum of these numbers including
1 divides 60, there can be no subgroup which is a union of conjugacy classes, and thereforeA 5 is a simple group.
We will demonstrate by induction that the alternating group \( A_n \) is simple for all \( n \geq 5 \) The base case begins with \( n = 5 \), and we assume \( n > 5 \) Let \( N \) be a non-trivial normal subgroup of \( A_n \) We examine the intersection \( N \cap A_{n-1} \), where \( A_{n-1} \) is the stabilizer in \( A_n \) of the point \( n \) This intersection is normal in \( A_{n-1} \) and, by the principle of induction, must be either the trivial group or \( A_{n-1} \).
A n − 1 In the second case,N A n − 1 , so contains all the elements of cycle type
In the context of group theory, it is evident that every even permutation can be expressed as a product of specific elements, leading to the conclusion that \( N = A_n \) Consequently, we can deduce that the intersection of \( N \) and \( A_{n-1} \) is trivial, implying that every non-identity element of \( N \) is distinct.
N is fixed-point-free (i.e fixes no points) Thus|N|n, for ifx, y∈N map the point 1 to the same point thenxy − 1 fixes 1 so is trivial.
The group A_n must include a non-trivial conjugacy class of elements, and it can be demonstrated that for n ≥ 5, no such class exists with fewer than n elements This verification is left as an exercise (Exercise 2.10) Consequently, this contradiction confirms that N cannot exist, establishing the simplicity of A_n An alternative proof of the simplicity of A_n is provided in Exercise 3.4.
Outer automorphisms
Automorphisms of alternating groups
If n4 thenA n has trivial centre, so thatA n ∼= Inn(A n )Aut(A n ) More- over, each element of S n induces an automorphism of A n , by conjugation in
The symmetric group S_n is isomorphic to a subgroup of Aut(A_n), and for n ≥ 7, it encompasses the entirety of Aut(A_n) This will be demonstrated in the following sections, with special thanks to Chris Parker for contributing to this proof.
First we observe that, since (a, b, c)(a, b, d) = (a, d)(b, c), the group A n is generated by its 3-cycles Indeed, it is generated by the 3-cycles (1,2,3),
The group \( A_n \) has no subgroup of index \( k \) less than \( n \), as the existence of such a subgroup would imply a homomorphism from \( A_n \) to a transitive subgroup of \( A_k \), which contradicts the simplicity of \( A_n \).
Lemma 2.2 If n7 andA n − 1 ∼=H A n , then H is the stabiliser of one of the npoints on whichA n acts.
The group H cannot act on a non-trivial orbit of length less than n−1; thus, if it is not a point stabilizer, it must act transitively on n points For n = 7, this is impossible since 7 does not divide the order of A6 For n > 8, each element of H corresponding to a 3-cycle of An−1 centralizes a subgroup isomorphic to An−4, which has at least n−4 points in its orbit Consequently, the 3-cycles of H can only move a maximum of four points, necessitating that they act as 3-cycles on all n points This is also applicable for n = 8, where the 3-cycles centralize A5, which includes C2 × C2, while elements of cycle type (3^2, 1^2) do not centralize.
The elements of H corresponding to (1,2,3) and (1,2,4) in A n − 1 generate a subgroup isomorphic to A 4, which maps to cycles (a, b, c) and (a, b, d) in A n Additionally, the elements corresponding to (1,2,j) map to (a, b, x), leading to the conclusion that the images (a, b, x) of the n−3 generating elements of H collectively move exactly n−1 points Consequently, H functions as one of the point stabilizers isomorphic to A n − 1.
Now we are ready to prove the theorem:
Any automorphism of the alternating group A_n permutes its subgroups, specifically the n subgroups that are isomorphic to A_{n-1} These subgroups correspond uniquely to the n points of the set Ω, which means that each automorphism can be viewed as a permutation of Ω Consequently, this shows that any automorphism is an element of the symmetric group S_n.
The theorem is also true forn= 5 and for n= 4 (see Exercise2.16).
The outer automorphism of S 6
The symmetric group S6 stands out due to its unique properties, particularly its exceptional outer automorphism This outer automorphism represents an isomorphism from S6 to itself that does not align with any permutation of the six underlying points Consequently, S6 can exert its influence on six points in an entirely distinct manner, highlighting its remarkable characteristics within the realm of symmetric groups.
To construct a non-inner automorphismφofS 6 we first note thatφmust map the point stabiliserS 5 to another subgroupH ∼=S 5 However,H cannot
fix any of the six points on which S 6 acts, so H must be transitive on these six points.
To establish a transitive action of the symmetric group S5 on six points, we can naturally achieve this through the conjugation action of S5 on its six Sylow 5-subgroups Alternatively, without invoking Sylow's theorems, we can note that the 24 elements of order 5 are organized into six cyclic subgroups, which are permuted transitively by the conjugation of elements in S5.
We have established a transitive subgroup H of index 6 within S6, allowing S6 to naturally and transitively act on the six cosets Hg through right multiplication This leads to the definition of a group homomorphism φ: S6 → Sym({Hg | g ∈ S6}) ∼= S6 The kernel of φ is trivial, as S6 lacks non-trivial normal subgroups of index 6 or greater, confirming that φ is a group isomorphism and, consequently, an automorphism of S6.
The automorphism φ is not classified as an inner automorphism since it transforms the transitive subgroup H into the stabilizer of the trivial coset H, which contradicts the property of inner automorphisms that maintain transitivity A more advanced version of this construction can be found in Section 3.3.5, specifically in the context of PSL2(5), while an alternative approach to the outer automorphism of S6 is discussed in Section 4.2.
The only outer automorphism of S6 is significant because S6 has an index of 2 in its full automorphism group Furthermore, it can be demonstrated that the outer automorphism group of A6 has an order of 4 Any automorphism that maps 3-cycles must map them to elements of order 3, which can be either 3-cycles or products of two disjoint 3-cycles Therefore, it is sufficient to establish that any automorphism mapping 3-cycles to 3-cycles belongs to S6, which can be proven using the arguments presented in Lemma 2.2 and Theorem 2.3.
Subgroups of S n
Intransitive subgroups
An intransitive subgroup H of S_n possesses two or more orbits on a set of n points If these orbits have lengths n_1, , n_r, then H is a subgroup of the group S_{n_1} × S_{n_r}, which includes all permutations that rearrange points within each orbit without intermixing them When r > 2, it is possible to permute all orbits except the first, resulting in a group S_{n_1} × S_{n_2} + + n_r that exists between H and S_n Consequently, in such cases, H cannot be considered a maximal subgroup.
If \( r = 2 \), the subgroup \( H = S_k \times S_{n-k} \) of \( S_n \) can be easily demonstrated to be a maximal subgroup, provided \( k = n - k \) Assuming \( k < n - k \), the factor \( S_k \) acts on \( \Omega_1 = \{1, 2, \ldots, k\} \) while \( S_{n-k} \) acts on \( \Omega_2 = \{k+1, \ldots, n\} \) For any permutation \( g \) not in \( H \), let \( K \) denote the subgroup generated by \( H \) and \( g \).
Our aim is to show thatKcontains all the transpositions ofS n , and therefore isS n
In the context of group theory, we observe that a subgroup \( H \) must map some points from \( \Omega_2 \) to \( \Omega_1 \), but not all, due to the cardinality condition \( |\Omega_2| > |\Omega_1| \) By selecting elements \( i \) and \( j \) from \( \Omega_2 \), where \( i_g \) belongs to \( \Omega_1 \) and \( j_g \) remains in \( \Omega_2 \), we establish that \( (i, j) \) is part of \( H \), leading to \( (i_g, j_g) \) being in \( H_g K \) By conjugating this transposition with elements of \( H \), we can generate all transpositions of \( S_n \) not already included in \( H \), resulting in \( K = S_n \) and confirming that \( H \) is a maximal subgroup of \( S_n \) This classification indicates that any other maximal subgroup must be transitive; for instance, the intransitive maximal subgroups of \( S_6 \) include \( S_5 \) and \( S_4 \times S_2 \).
Transitive imprimitive subgroups
In the case whenk=n−k, this proof breaks down, and in fact the subgroup
S k ×S k is not maximal in S 2k This is because there is an element h in
S 2k which interchanges the two orbits of sizek, and normalises the subgroup
The wreath product of the symmetric group S_k with S_2 forms a maximal subgroup of S_2k This relationship can be demonstrated using a method similar to that outlined in Exercise 2.27.
The partitioning of a set of n points into m subsets of equal size k, where n = km, allows the wreath product S_k ≀ S_m to act on this partition In this structure, the base group S_k operates by permuting each of the m subsets independently, while the wreathing action of S_m permutes the m orbits created by the base group Notably, this subgroup is maximal within S_n, leading to a comprehensive list of all transitive imprimitive maximal subgroups of S_n, specifically those of the form S_k S_m, where both k and m are greater than 1 and n equals km.
S 6areS 2 S 3 (preserving a set of three blocks of size 2, for example generated by the three permutations (1,2), (1,3,5)(2,4,6) and (3,5)(4,6)) andS 3 S 2
(preserving a set of two blocks of sise 3, for example generated by the three permutations (1,2,3), (1,2) and (1,4)(2,5)(3,6)).
Primitive wreath products
We have fully classified the imprimitive maximal subgroups of \( S_n \), indicating that all remaining maximal subgroups must be primitive For instance, when \( n = k^2 \), arranging the \( n \) points in a \( k \times k \) array allows one copy of \( S_k \) to permute the columns while fixing each row, and another copy to permute the rows while fixing each column These two copies of \( S_k \) commute, generating a group \( H \cong S_k \times S_k \), which is imprimitive due to the rows and columns forming separate systems of imprimitivity However, by adding the permutation that reflects across the main diagonal, mapping rows to columns, we form the group \( S_k S_2 \), which is primitive For example, the primitive subgroup \( S_3 S_2 \) exists within \( S_9 \) but is not maximal.
The smallest maximal case is represented by the subgroup S5 within S25 By generalizing this concept to an m-dimensional array where n equals k times m, with k greater than 2 and m greater than 1, we derive a primitive action of the group.
S k S m on k m points To make this more explicit, we identify Ω with the Cartesian productΩ 1 m of mcopies of a setΩ 1 of sizek, and let an element
(π 1 , , π m ) of the base groupS k m act by (a 1 , , a m )→(a 1 π 1 , , a m π m ) (2.7) for alla i ∈Ω 1 The wreathing action ofρ − 1 ∈S m is then given by the natural action permuting the coordinates, thus: ρ − 1 : (a 1 , , a m )→(a 1 ρ , , a m ρ ) (2.8)
The action of the wreath product is often referred to as the product action, differentiating it from the imprimitive action on k points discussed in Section 2.5.2 We will not be proving the maximality of these subgroups within S_n.
A n , although they are in fact maximal inA n if k5 and k m − 1 is divisible by 4, and maximal inS n ifk5 andk m − 1 is not divisible by 4.
Affine subgroups
The affine groups are essentially the symmetry groups of vector spaces Letp be a prime, and let F p =Z/pZdenote the field of orderp(for more on finite
The vector space \( V \) of \( k \)-tuples of elements from the finite field \( F_p \) contains \( p^k \) elements and is associated with a symmetry group known as the affine general linear group, denoted as \( AGL_k(p) \) This group is a semidirect product of the translation group \( t_a: v \rightarrow v + a \) and the general linear group \( GL_k(p) \), which consists of all invertible \( k \times k \) matrices over \( F_p \) Acting as permutations of the vectors, \( AGL_k(p) \) is a subgroup of the symmetric group \( S_n \), where \( n = p^k \) The translation subgroup is normal and isomorphic to the additive group of the vector space, which can be represented as a direct product of \( k \) copies of the cyclic group \( C_p \) Thus, it forms an elementary abelian group of order \( p^k \), denoted \( E_{p^k} \) or simply \( p^k \) In summary, we have the relation \( AGL_k(p) \cong p^k : GL_k(p) \).
An example of an affine group is AGL3(2) ∼= 2 3:GL3(2), which acts as a permutation group on the eight vectors of F2 3 and embeds in S 8, with all its elements being even permutations, thus embedding in A 8 Another example is AGL1(7) ∼= 7:6, a maximal subgroup of S 7, although its intersection with A 7 forms the group 7:3, which is not maximal in A 7 Both groups, 7:6 and 7:3, are examples of Frobenius groups, characterized as transitive non-regular permutation groups with a trivial stabilizer for any two points Additionally, dihedral groups D 2n ∼= n:2, representing the symmetries of a regular n-gon, also serve as examples of Frobenius groups.
Subgroups of diagonal type
The diagonal type groups are less easy to describe They are built from a non-abelian simple groupT, and have the shape
The normal subgroup TS k is extended by a group of outer automorphisms that uniformly act on all k copies of T This structure includes a subgroup Aut(T)×S k, which consists of a diagonal copy of T formed by all elements (t, , t) where t belongs to T, alongside its outer automorphism group and the permutation group With an index of |T| k − 1, the permutation action on the cosets of this subgroup facilitates an embedding of the entire group into S n, where n equals |T| k − 1.
The smallest example of such a group is (A 5 ×A 5 ):(C 2 ×C 2 ) acting on the cosets of a subgroup S 5 ×C 2 This group is the semidirect product of
The group \( A_5 \times A_5 \) can be represented as \( \{(g, h) | g, h \in A_5\} \), and is influenced by the automorphism group \( C_2 \times C_2 \) generated by the mappings \( \alpha: (g, h) \rightarrow (g \pi, h \pi) \) and \( \beta: (g, h) \rightarrow (h, g) \), where \( \pi \) is the transposition \( (1,2) \) The point stabilizer, which is the centralizer of \( \beta \), is generated by \( \alpha \), \( \beta \), and the set \( \{(g, g) | g \in A_5\} \) Consequently, the group's action on 60 points can alternatively be described as the action by conjugation on the 60 conjugates of \( \beta \).
Almost simple groups
Almost simple groups, denoted as G, are defined by their relation to a simple group T, satisfying the condition GAut(T) These groups may include a simple group along with some or all of its outer automorphism group When considering any maximal subgroup M of G, the action of G on the cosets of M is primitive, allowing G to be embedded as a primitive subgroup of S n, where n represents the index |G : M| The nature of almost simple maximal subgroups within S n is generally chaotic, and fully characterizing them necessitates an exhaustive understanding of the maximal subgroups of all almost simple groups, presenting a challenging and complex problem.
According to the findings of Liebeck, Praeger, and Saxl, every embedding of GinS n is considered maximal under specific technical conditions, except for those listed as exceptions Additionally, it is established that as n approaches infinity, almost all values of n lack almost simple maximal subgroups within S n or A n.