Handbook of mathematics for engineers and scienteists part 38 potx

5.8. SOME FACTS FROM GROUP THEORY 227 T HEOREM. The homomorphic image f(G) is a group. The image f(e) of the identity element e G is the identity element of the group f(G) . Mutually inverse elements of G correspond to mutually inverse images in f(G) . Two groups G 1 and G 2 are said to be isomorphic if there exists a one-to-one mapping f of G 1 onto G 2 such that f (ab)=f (a)f (b)foralla, b G 1 . Such a mapping is called an isomorphism or isomorphic mapping of the group G 1 onto the group G 2 . T HEOREM. Any isomorphism of groups is invertible, and the inverse mapping is also an isomorphism. An isomorphic mapping of a group G onto itself is called an automorphism of G.If f 1 : G → G and f 2 : G → G are two automorphisms of a group G, one can define another automorphism f 1 ◦ f 2 : G → G by letting (f 1 ◦ f 2 )(g)=f 1 (f 2 (g)) for all g G.This automorphism is called the composition of f 1 and f 2 , and with this composition law, the set of all automorphisms of G becomes a group called the automorphism group of G. 5.8.1-4. Subgroups. Cosets. Normal subgroups. Let G be a group. A subset G 1 of the group G is called a subgroup if the following conditions hold: 1. For any a and b belonging to G 1 , the product ab belongs to G 1 . 2. For any a belonging to G 1 , its inverse a –1 belongs to G 1 . These conditions ensure that any subgroup of a group is itself a group. Example 5. The identity element of a group is a subgroup. The subset of all even numbers is a subgroup of the additive group of all integers. The product of two subsets H 1 and H 2 of a group G is a set H 3 that consists of all elements of the form h 1 h 2 ,whereh 1 H 1 , h 2 H 2 . In this case, one writes H 3 = H 1 H 2 . Let H be a subgroup of a group G and a some fixed element of G.ThesetaH is called a left coset,andthesetHa is called a right coset of the subgroup H in G. Properties of left cosets (right cosets have similar properties): 1. If a H,thenaH ≡ H. 2. Cosets aH and bH coincide if a –1 b H. 3. Two cosets of the same subgroup H either coincide or have no common elements. 4. If aH is a coset, then a aH. A subgroup H of a group G is called a normal subgroup of G if H = a –1 Ha for any a G. This is equivalent to the condition that aH = Ha for any a G, i.e., every right coset is a left coset. 5.8.1-5. Factor groups. Let H be a normal subgroup of a group G. Then the product of two cosets aH and bH (as subsets of G)isthecosetabH. Consider the set Q whose elements are cosets of the subgroup H in G, and define the product of the elements of Q as the product of cosets. Endowed with this product, Q becomes a group, denoted by Q = G/H and called the quotient group of G with respect to the normal subgroup H. The mapping f : G → G/H that maps each a G to the corresponding coset aH is a homomorphism of G onto G/H. If f : G → G is a homomorphism of groups, the set of all elements of G mapped into the identity element of G is called the kernel of f and is denoted by ker f = {g G : f(g)=f (e)}. 228 ALGEBRA THEOREM 1. If f is a homomorphism of a group G onto a group G and H is the set of all elements of G that are mapped to f(e) ( e is the identity element of G ), then H is a normal subgroup in G . THEOREM 2(ON GROUP HOMOMORPHISMS). If f is a homomorphism of a group G onto a group G and H is the normal subgroup of G consisting of the elements mapped to the identity element of G , then the group G and the quotient group G/H are isomorphic. Thus, given a homomorphism f of a group G onto a group  G,thekernelH of the homomorphism is a normal subgroup of G, and conversely any normal subgroup H of G is the kernel of the homomorphism of G onto the quotient group G/H. Remark. Given a homomorphism of a group G onto a set G, all elements of the group G are divided into mutually disjoint classes, each class containing all elements of G that are mapped into the same element of G. Example 6. Let R n be the n-dimensional linear coordinate space, which is an abelian group with respect to addition of its elements. This space is the direct product of one-dimensional spaces: R n = R 1 (1) ⊗ ···⊗ R 1 (n) . Since R 1 (n) is an abelian subgroup, the set R 1 (n) is a normal subgroup of the group R n . The coset corresponding to an element a R n is the straight line passing through a in the direction parallel to the straight line R 1 (n) ,and the quotient group R n /R 1 (n) is isomorphic to the (n – 1)-dimensional space R n–1 : R n–1 = R n /R 1 (n) = R 1 (1) ⊗ ···⊗ R 1 (n–1) . 5.8.2. Transformation Groups 5.8.2-1. Group of linear transformations. Its subgroups. Let V bearealfinite-dimensional linear space and let A : V → V be a nondegenerate linear operator. This operator can be regarded as a nondegenerate linear transformation of the space V ,sinceA maps different elements of V into different elements, and for any y V there is a unique x V such that Ax = y. The set of all nondegenerate linear transformations A of the n-dimensional real linear space V is denoted by GL(n). The product AB of linear transformations A and B in GL(n)isdefined by the relation (AB)x = A(Bx)forallx V . This product is a composition law on GL(n). T HEOREM. The set GL(n) of nondegenerate linear transformations of an n -dimensional real linear space V with the above product is a group. The group GL(n) is called the general linear group of dimension n. A subset of GL(n) consisting of all linear transformations A such that det A = 1 is a subgroup of GL(n) called the special linear group of dimension n and denoted by SL(n). A sequence {A k } of elements of GL(n)issaidtobeconvergent to an element A GL(n) as k →∞if the sequence {A k x} converges to Ax for any x V . Types of subgroups of GL(n): 1. Finite subgroups are subgroups with finitely many elements. 2. Discrete subgroups are subgroups with countably many elements. 3. Continuous subgroups are subgroups with uncountably many elements. 5.8. SOME FACTS FROM GROUP THEORY 229 Example 1. The subgroup of reflections with respect to the origin is finite and consists of two elements: the identity transformation and the reflection x → –x. The subgroup of rotations of a plane with respect to the origin by the angles kϕ (k = 0, 1, 2, and ϕ is a fixed angle incommensurable with π) is a discrete subgroup. The subgroups of all rotations of a three-dimensional space about a fixed axis are a continuous subgroup. A continuous subgroup of GL(n)issaidtobecompact if from any infinite sequence of its elements one can extract a subsequence convergent to some element of the subgroup. 5.8.2-2. Group of orthogonal transformations. Its subgroups. Consider the set O(n) that consists of all orthogonal transformations P of the n-dimensional Euclidean space V , i.e., P T P = PP T = I (see Paragraph 5.2.3-3 and Section 5.4). This set is a subgroup of GL(n) called the orthogonal group of dimension n. All orthogonal transformations are divided into two classes: 1. Proper orthogonal transformations,forwhichdetP =+1. 2. Improper orthogonal transformations,forwhichdetP =–1. The set of proper orthogonal transformations forms a group called the special orthogonal group of dimension n and denoted by SO(n). In the two-dimensional orthogonal group O(2) there is a subgroup of rotations by the angles kϕ,wherek = 0, 1, 2, and ϕ is fixed. If a k is its element corresponding to k and a = a 1 , then the element a k (k > 0)hastheform a k = a ⋅ a ⋅ ⋅ a    k times = a k (k = 1, 2, 3, ). Denoting by a –1 the inverse of a = a 1 , and the identity element by a 0 , we see that each element of this group has the form a k = a k (k = 0, 1, 2, ). Groups whose elements admit such a representation in terms of a single element are said to be cyclic. Such groups are discrete. There are two cyclic groups of rotations (p and q are coprime numbers): 1. If ϕ ≠ 2πp/q (i.e., the angle ϕ is incommensurable with π), then all elements are distinct. 2. If ϕ = 2πp/q,thena k+q = a k (a q = a 0 ). Such groups are called cyclic groups of order q. Consider groups of mirror symmetry. Each of them consists of two elements: the identity element and a reflection with respect to the origin. Let {I, P} be a subgroup of O(3) consisting of the identity I and the reflection P of the three-dimensional space with respect to the origin, Px =–x. This is an improper subgroup. It is isomorphic to the group Z 2 of residues modulo 2. The subgroup {I, P} is a normal subgroup in O(3), and the subgroup SO(3) (consisting of proper orthogonal transformations) is isomorphic to the quotient group O(3)/{I, P}. 5.8.2-3. Unitary groups. By analogy with Paragraph 5.8.2-2, one can consider groups of linear transformations of a complex linear space. In the general linear group of transformations of a unitary space, one considers unitary groups U (n), which are analogues of orthogonal groups. In the group U(n)ofunitary transformations, one considers the subgroup SU(n) that consists of unitary transformations whose determinant is equal to 1. 230 ALGEBRA 5.8.3. Group Representations 5.8.3-1. Linear representations of groups. Terminology. A linear representation of a group G in the finite-dimensional Euclidean space V n is a homomorphism of G to the group of nondegenerate linear transformations of V n ;inother words, a linear representation of G is a mapping D that associates each element a G with a nondegenerate linear transformation D(a)ofthespaceV n , so that for any a 1 and a 2 in G, we have D(a 1 a 2 )=D(a 1 )D(a 2 ). Thus, for any g G, its image D(g) is an element of the group GL(n), and the set D(G) consisting of all transformations D(g), g G, is a subgroup of GL(n) isomorphic to the quotient group G/ker D,wherekerD is the kernel of the homomorphism D, i.e., the set of all g such that D(g) is the identity element of the group GL(n). The subgroup D(G) is often also called a representation of the group G. The space V n is called the representation space; n is called the dimension of the representation; and the basis in V n is called the representation basis. The trivial representation of a group is its homomorphic mapping onto the identity element of the group GL(n). A faithful representation of a group G is an isomorphism of G onto a subgroup of GL(n). 5.8.3-2. Matrices of linear representations. Equivalent representations. If D (μ) (G) is a representation of a group G, each g G corresponds to alinear transformation D (μ) (g), whose matrix in the basis of the representation D (μ) (G) is denoted by [D (μ) ij (g)]. Two representations D (μ 1 ) (G)andD (μ 2 ) (G) of a group G in the same space E n are said to be equivalent if there exists a nondegenerate linear transformation C of the space E n such that D (μ 1 ) (g)=C –1 D (μ 2 ) (g)C for each g G. The choice of a basis in the representation space is important, since the matrices corresponding to the group elements may have some standard fairly simple form inthat basis, and this allows one to make important conclusions with regards to a given representation. 5.8.3-3. Reducible and irreducible representations. A subspace V  of V n is called invariant for a representation D(G) if it is invariant with respect to each linear operator in D(G). Suppose that all matrices of some three-dimensional representation D(G) have the form  A 1 A 2 OA 3  , A 1 ≡  a 11 a 12 a 21 a 22  , A 2 ≡  a 13 a 23  , A 3 ≡ ( a 33 ), O ≡ ( 00). The product of such matrices has the form (see Paragraph 5.2.1-10)  A  1 A  2 OA  3  A  1 A  2 OA  3  =  A  1 A  1 A  2 OA  3 A  3  , and therefore the structure of the matrices is preserved. Thus, the matrices A 1 form a two-dimensional representation of the given group G and the matrices A 3 form its one- dimensional representation. In such cases, one says that D(G)isareducible representation. 5.8. SOME FACTS FROM GROUP THEORY 231 If all matrices of a representation have the form of size n × n  A 1 O OA 2  , then the square matrices A 1 and A 2 form representations, the sum of their dimensions being equal to n. In this case, the representation is said to be completely reducible.The representation induced on an invariant space by a given representation D(G) is called a part of the representation D(G). A representation D(G) of a group G is said to be irreducible if it has only two invariant subspaces, V n and O. Otherwise, it is said to be reducible. Any representation can be expressed in terms of irreducible representations. 5.8.3-4. Characters. Let D(G)beann-dimensional representation of a group G,andlet[D ij (g)] be the matrix of the operator corresponding to the element g G.Thecharacter of an element g G in the representation D(G)isdefined by χ(g)= n  i=1 D ii (g) = Tr([D ij (g)]). Thus, the character of an element does not depend on the representation basis and is, therefore, an invariant quantity. An element b G is said to be conjugate to the element a G if there exists u G such that uau –1 = b. Properties of conjugate elements: 1. Any element is conjugate to itself. 2. If b is conjugate to a,thena is conjugate to b. 3. If b is conjugate to a and c is conjugate to b,thenc is conjugate to a. The characters of all elements belonging to one and the same class of conjugate elements coincide. The characters of elements for equivalent representations coincide. 5.8.3-5. Examples of group representations. 1 ◦ .LetG be a group of symmetry of three-dimensional space consisting of two elements: the identity transformation I and the reflection P with respect to the origin, G = {I, P}. The multiplication of elements of this group is described by the table I P I I P P P I 1. One-dimensional representation of the group G. In the space E 1 , we chose a basis e 1 and consider the matrix A (1) of the nondegenerate transformation A 1 of this space: A (1) =(1). The transformation A 1 forms a subgroup in the 232 ALGEBRA group GL(1) of all linear transformations of E 1 , and the multiplication in this subgroup is described by the table A (1) A (1) A (1) We obtain a one-dimensional representation D (1) (G) of the group G by letting D (1) (I)=A (1) , D (1) (P)=A (1) . These relations define a homomorphism of the group G to GL(1) and thus define its representation. 2. A two-dimensional representation of the group G. In E 2 , we choose a basis e 1 , e 2 and consider the matrices A (2) , B (2) of linear transfor- mations A 2 , B 2 of this space: A (2) =  1 0 0 1  , B (2) =  0 1 1 0  . The transformations A 2 , B 2 form a subgroup in the group GL(2) of linear transformations of E 2 . The multiplication in this subgroup is defined by the table A (2) B (2) A (2) A (2) B (2) B (2) B (2) A (2) We obtain a two-dimensional representation D (2) (G) of the group G by letting D (2) (I)=A (2) , D (2) (P)=B (2) . These relations define an isomorphism of G onto the subgroup {A (2) , B (2) } of GL(2), and therefore define its representation. 3. A three-dimensional representation of the group G. Consider the linear transformation A (3) of E 3 definedbythematrix A (3) =  100 010 001  . This transformation forms a subgroup in GL(3) with the multiplication law A (3) A (3) = A (3) . One obtains a three-dimensional representation D (3) (G) of the group G by letting D (3) (I)= A (3) , D (3) (P)=A (3) . 4. A four-dimensional representation of the group G. Consider linear transformations A (4) and B (4) of E 4 definedbythematrices A (4) = ⎛ ⎜ ⎝ 1 0 00 0 1 00 00 1 0 00 0 1 ⎞ ⎟ ⎠ , B (4) = ⎛ ⎜ ⎝ 0 1 00 1 0 00 00 0 1 00 1 0 ⎞ ⎟ ⎠ . The transformations A (4) and B (4) form a subgroup in GL(4) with the multiplication defined by a table similar to that in the two-dimensional case. One obtains a four-dimensional representation D (4) (G) of the group G by letting D (4) (I)=A (4) , D (4) (P)=A (B) . Remark. The matrices A (4) and B (4) maybewrittenintheformA (4) =  A (2) 0 0 A (2)  , B (4) =  B (2) 0 0 B (2)  , and therefore the representation D (4) (G) is sometimes denoted by D (4) (G)=D (2) (G)+D (2) (G)=2D (2) (G). In a similar way, one may use the notation D (3) (G)=3D (1) (G). In this way, one can construct representations of the group G of arbitrary dimension. 2 ◦ . The symmetry group G = {I, P} for the three-dimensional space is a normal subgroup of the group O(3). The subgroup SO(3) ⊂ O(3) formed by proper orthogonal transformations is isomorphic to the quotient group O(3)/{I, P}. REFERENCES FOR CHAPTER 5 233 Since any group admits a homomorphic mapping onto its quotient group, there is a homomorphism of the group O(3) onto SO(3). This homomorphism is defined as follows: if a is a proper orthogonal transformation in O(3), its image in SO(3) coincides with a;andifa  is an improper orthogonal transformation, its image is the proper orthogonal transformation Pa  . In this way, one obtains a three-dimensional representation DO(3) of the group of orthogonal transformations O(3) in terms of the group SO(3) of proper orthogonal trans- formations. References for Chapter 5 Anton, H., Elementary Linear Algebra, 8th Edition, Wiley, New York, 2000. Barnett, S., Matrices in Control Theory, Rev. Edition, Krieger Publishing, Malabar, 1985. Beecher, J. A., Penna, J. A., and Bittinger, M. L., College Algebra, 2nd Edition, Addison Wesley, Boston, 2004. Bellman, R. E., Introduction to Matrix Analysis (McGraw-Hill Series in Matrix Theory), McGraw-Hill, New York, 1960. Bernstein, D. S., Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, 2005. Blitzer,R.F.,College Algebra, 3rd Edition, Prentice Hall, Englewood Cliffs, New Jersey, 2003. Bronson, R., Schaum’s Outline of Matrix Operations, McGraw-Hill, New York, 1988. Cullen, C. G., Matrices and Linear Transformations, 2nd Edition, Dover Publications, New York, 1990. Davis, P. J. (Editor), The Mathematics of Matrices: A First Book of Matrix Theory and Linear Algebra, 2nd Edition, Xerox College Publications, Lexington, 1998. Demmel, J. W., Applied Numerical Linear Algebra, Society for Industrial & Applied Mathematics, University City Science Center, Philadelphia, 1997. Dixon,J.D.,Problems in Group Theory, Dover Publications, New York, 1973. Dugopolski, M., College Algebra, 3rd Edition, Addison Wesley, Boston, 2002. Eves, H., Elementary Matrix Theory, Dover Publications, New York, 1980. Franklin, J. N., Matrix Theory, Dover Publications, New York, 2000. Gantmacher, F. R., Matrix Theory, Vol. 1, 2nd Edition, American Mathematical Society, Providence, Rhode Island, 1990. Gantmacher, F. R., Applications of the Theory of Matrices, Dover Publications, New York, 2005. Gelfand, I. M., Lectures on Linear Algebra, Dover Publications, New York, 1989. Gelfand, I. M. and Shen, A., Algebra, Birkhauser, Boston, 2003. Gilbert, J. and Gilbert, L., Linear Algebra and Matrix Theory, 2nd Edition, Brooks Cole, Stamford, 2004. Golub, G. H. and Van Loan, C. F., The Matrix Computations, 3nd Edition (Johns Hopkins Studies in Mathematical Sciences), The Johns Hopkins University Press, Baltimore, Maryland, 1996. Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 1, North Holland, Amsterdam, 1996. Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 2, North Holland, Amsterdam, 2000. Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 3, North Holland, Amsterdam, 2003. Horn,R.A.andJohnson,C.R.,Matrix Analysis, Rep. Edition, Cambridge University Press, Cambridge, 1990. Horn,R.A.andJohnson,C.R.,Topics in Matrix Analysis, New Ed. Edition, Cambridge University Press, Cambridge, 1994. Householder, A. S., The Theory of Matrices in Numerical Analysis (Dover Books on Mathematics), Dover Publications, New York, 2006. Jacob, B., Linear Functions and Matrix Theory (Textbooks in Mathematical Sciences), Springer, New York, 1995. Kostrikin, A. I. and Artamonov, V. A. (Editors), Exercises in Algebra: A Collection of Exercises in Algebra, Linear Algebra and Geometry (Expanded), Gordon & Breach, New York, 1996. Kostrikin, A. I. and Manin, Yu. I., Linear Algebra and Geometry, Gordon & Breach, New York, 1997. Kostrikin, A. I. and Shafarevich, I. R. (Editor), Algebra I: Basic Notions of Algebra , Springer-Verlag, Berlin, 1990. Kurosh,A.G.,Lectures on General Algebra, Chelsea Publishing, New York, 1965. Kurosh,A.G.,Algebraic Equations of Arbitrary Degrees, Firebird Publishing, New York, 1977. Kurosh,A.G.,Theory of Groups, Chelsea Publishing, New York, 1979. Lancaster, P. and Tismenetsky, M., The Theory of Matrices, Second Edition: With Applications (Computer Science and Scientific Computing), Academic Press, Boston, 1985. . Baltimore, Maryland, 1996. Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 1, North Holland, Amsterdam, 1996. Hazewinkel, M. (Editor), Handbook of Algebra, Vol. 2, North Holland, Amsterdam,. product of two cosets aH and bH (as subsets of G)isthecosetabH. Consider the set Q whose elements are cosets of the subgroup H in G, and define the product of the elements of Q as the product of cosets. Endowed. f 2 )(g)=f 1 (f 2 (g)) for all g G.This automorphism is called the composition of f 1 and f 2 , and with this composition law, the set of all automorphisms of G becomes a group called the automorphism group of G. 5.8.1-4.