FINITE GROUP THEORY. FOR COMBINATORISTS Volume one. Jin Ho Kwak. Department of Mathematics POSTECH Pohang, Korea


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1 FINITE GROUP THEORY FOR COMBINATORISTS Volume one Jin Ho Kwak Department of Mathematics POSTECH Pohang, Korea Ming Yao Xu Department of Mathematics Peking University Beijing , P.R. China This work is supported by Com 2 MaCKOSEF, Korea.
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3 Preface This book is designed for a text on finite group theory for combinatorists, mainly for graphtheorists. It has three parts. This volume is the first part which gives the basic concepts and theorems on abstract groups. The volume 2 contains the second part which is a text and also a reference book on permutation groups. The volume 3 contains the third part which gives some examples for applying group theory to the combinatorial problems. For volume 1, the second author used it three times as a text on group theory for graduate students. The first one was in 2000 at a summer school for graduate students held in Kunming, China. The second time was in spring semester in 2001 at Temple University, the United States. The third time was in spring semester in 2005 at Pohang University of Science and Technology, Korea. The present version is a expansion of his lecture notes. We would take this opportunity to thank all people who give us valuable comments and suggestions for revising this book, including many students. Especially we are grateful to Professor Seymour Lipschutz, at Temple University, who read the manuscript carefully and corrected some errors and polished the English. We also thank Shanxi Teachers University, China, for its hospitality; part of work of writing and revising this book was done when the second author visited there. Finally we are grateful to the Combinatorial and Computational Mathematics Center at Pohang University of Science and Technology, Korea, for accepting this book as one of the Lecture Note Series. iii Jin Ho Kwak MingYao Xu July 2005, in Pohang, Korea
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5 Contents Preface iii 1 Basic concepts Groups and subgroups Normal subgroups and direct products Examples of groups Commutators and solvable groups Automorphisms Some miscellaneous properties Free groups, generators and relations Graphs of groups and Groups of graphs Group actions Group actions on a set Sylow Theorems pgroups and solvable groups Transitive permutation representations Transfer and Burnside Theorem The structure theory The JordanHölder Theorem Decomposition into direct products Extensions and SchurZassenhaus Theorem Classifying groups of special orders Wreath products v
6 vi 4 Nilpotent groups and solvable groups Commutators Nilpotent groups Frattini subgroups pgroups Hall s enumeration principle Solvable groups Fitting subgroups Sylow basis for solvable groups Representations of finite groups Representations of groups Characters of groups Induced representations Applications Selected Answers and Hints 221 Bibliography 247 Index 255
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8 viii Notation Notation N, Z natural numbers and integers Q, R, C rational, real and complex numbers (m, n) greatest common divisor of m and n G, H,... sets, groups, etc X, D,... classes of groups α, β, γ,... functions x, y, z,... elements of a set x α image of x under α G α image of G under α Ker α kernel of α x y y 1 xy Cl(g) the conjugacy class containing g [x, y] x 1 y 1 xy H = G H is isomorphic with G H char G H is a characteristic subgroup of G H G H is isomorphic to a subgroup of G H G, H < G H is a subgroup, a proper subgroup of the group G H G, H G H is a normal subgroup, a proper normal subgroup of G H sn G H is a subnormal subgroup of G H 1 H 2 H n product of subsets of a group X λ λ Λ subgroup generated by subsets X λ of a group X R group presented by generators X and relators R M subgroup generated by M N 1, N 2,..., N s subgroup generated by N 1, N 2,..., N s d(g) minimum number of generators of G G n subgroup generated by all g n where g G S cardinality of the set S G : H index of the subgroup H in the group G o(x) order of the group element x
9 Notation ix C G (H), N G (H) H G H g H G or Core G (H) Aut(G), Inn(G) Out(G) Hol(G) Hom Ω (G, H) End Ω (G) H 1 H n, H 1 H n N H H K H K G = [G, G] G (i) G i, Z i (G) k(g) Z(G) F (G) Φ(G) M(G) O π (G) l π (G) S X S n, A n D 2n Q 2 n Z n R RG GL(V ) GL(n, F), SL(n, F) PGL(n, F), PSL(n, F) centralizer, normalizer of H in G normal closure of H in G conjugate of H by g core of M in G automorphism group, inner automorphism group of G Aut(G)/Inn(G), outer automorphism group of G holomorph of G set of Ωhomomorphisms from G to H set of Ωendomorphisms of G set product direct products, direct sums semidirect products wreath products tensor product derived subgroup of a group G term of the derived series of G terms of the lower central series, the upper central series of G the number of conjugacy classes of G center of G Fitting subgroup of G Frattini subgroup of G Schur multiplicator of G maximal normal πsubgroup of G πlength of G symmetric group on X symmetric, alternating groups of degree n dihedral group of order n generalized quaternion group of order 2 n Z/nZ, cyclic group of order n group of units of a ring R with identity group ring of a group G over a ring R with identity element group of nonsingular linear transformations of a vector space V general linear and special linear linear groups projective general linear and projective special linear groups
10 x Notation M G, χ G max, min E ij F q, GF (q) F d, V (n, F) AG(d, F), AG(d, q) PG(d, F), PG(d, q) S(t, k, v) S n, A n GL(d, F), SL(d, F), ΓL(d, F) GL(d, q), SL(d, q), ΓL(d, q) AGL(d, F), ASL(d, F), AΓL(d, F) AGL(d, q), ASL(d, q), AΓL(d, q) PGL(d, F), PSL(d, F), PΓL(d, F) PGL(d, q), PSL(d, q), PΓL(d, q) Sp(2m, F) Sp(2m, q) PGU(d, F), PSU(d, F), PΓU(d, F) PGU(d, q), PSU(d, q), PΓU(d, q) Sz(2 s ) and R(3 s ) M 10,..., M 24 W 10,..., W 24 fix(x), supp(x) Ω {k}, Ω (k) x N G (H) G H, G m G.H G : H induced module, induced character maximal, minimal conditions matrix with (i, j) entry 1 and other entries 0 field with q elements vector space of dimension n over F affine geometry over F and over F q projective geometry over F and over F q Steiner system symmetric and alternating groups of degree n linear groups over F linear groups over GF (q) affine groups over F affine groups over GF (q) projective groups over F projective groups over GF (q) sympletic groups over F sympletic groups over GF (q) unitary groups over F unitary groups over GF (q) Suzuki and Ree groups Mathieu groups Witt geometries set of fixed points and support of x sets of ksubsets and ktuples from Ω largest integer x socle of G direct product, direct power an extension of G by H a split extension of G by H
11 Volume one Abstract Groups
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13 Chapter 1 Basic concepts We begin with some basic definitions and elementary properties of finite groups, up to but not including Sylow s Theorems. Most of the material in this chapter can be found in lots of textbooks of abstract algebra; so we omit a proof in many cases. 1.1 Groups and subgroups Definition A nonempty set G with a closed binary operation called a multiplication is called a monoid if the following two axioms hold: (1) Associative law: (ab)c = a(bc) for all a, b, c G; (2) Identity element: there exists 1 G such that 1a = a1 = a for all a G. A monoid is called a group if (3) Inverse element: for each a G, there is an element a 1 G such that aa 1 = a 1 a = 1. A group G is called abelian (or commutative) if (4) Commutative law: ab = ba for all a, b G. We use 1 to denote both the identity element in a group and the group consisting of the identity element only. We use G to denote the 3
14 4 Basic concepts cardinality of a set G, and if G is a group we call G the order of G. The group G is called finite if G is a positive integer; otherwise G is called infinite. Elementary examples of groups come from number systems. The complex number system C is a group under ordinary addition; but not a group under ordinary multiplication because 0 has no inverse. However, the nonzero complex numbers C \ {0} form a group under ordinary multiplication. As subsets, the integers Z, the rational numbers Q, and the real numbers R are groups under addition, and the nonzeros of Q and R form groups under multiplication but Z \ {0} is not. Note that all mentioned groups are abelian. For another example, let M = {1, 2,..., n} and let S M be the set of all bijections M M, called permutations. Under the usual composition of functions, 1 S M becomes a group, called the symmetric group on n elements and denoted S n. Note that S n = n! and S n is not abelian for n 3. Throughout this book, all groups are assumed to be finite unless otherwise stated. By the associative law, products of any finite number of elements of G in a certain order are meaningful. Thus one may define the powers of an element a of G as follows: For any positive integer n, Obviously we have a n = aa } {{ a}, a 0 = 1, a n = (a 1 ) n. n a m a n = a m+n and (a m ) n = a mn for any integers m, n. Let H and K be subsets of a group G. We define a product HK := {hk h H, k K}. (1.1) 1 For any σ, τ S M, one can define their multiplication σ τ as either σ τ or τ σ, where (σ τ)(x) = σ(τ(x)). In the former case, we write (σ τ)(x) = (σ τ)(x) like the usual function notation. However, in the latter case, we use x σ instead of the usual notation σ(x) to get x σ τ = (x σ ) τ. Throughout this book, the latter case is preferred.
15 Groups and subgroups 5 Since G is associative, so is this product operation. If K = {a} then we write Ha for H{a}; similarly we write ah for {a}h. Also we define H 1 = {h 1 h H}; and for any positive integer n, define H n = {h 1 h 2 h n h i H}. Definition A nonempty subset H of a group G is called a subgroup of G, denoted by H G, if H 2 H and H 1 H. It is easy to see that if H is a subgroup of G then H 2 = H, H 1 = H and 1 H, and hence H satisfies the axioms of a group. Obviously, every group G 1 has at least two subgroups, 1 and G itself; 1 is called the trivial subgroup of G. Sometimes, we write H < G if H G but H G. Proposition Let G be a group and H G a nonempty subset. Then the following statements are equivalent: (1) H G; (2) ab H and a 1 H for any a, b H; (3) ab 1 H (or a 1 b H) for any a, b H. (4) H 2 H (if G is finite). The intersection of a collection of subgroups of a group G is also a subgroup of G, but the union of several subgroups is not necessarily a subgroup. Definition Let G be a group and M G. The intersection of all subgroups of G containing M is called the subgroup generated by M, denoted by M. It is easy to see that M = {1, a 1 a 2 a n a i M M 1, n = 1, 2,...}.
16 6 Basic concepts If M = G, M is said to be a generating set of G or G is generated by M. A group generated by one element is called a cyclic group. A group generated by a finite number of elements is called a finitely generated group. For an element a in a group G, we call the order of the subgroup a the order of a, denoted by o(a), that is o(a) = a. One can show that o(a) is the smallest positive integer n satisfying a n = 1; and if such an integer n does not exist we say o(a) =. For a group G, we call the least common multiple of the orders of elements of G the exponent of G, denoted by exp G. Note that for any two subgroups H and K of a group G, their product HK is not necessary to be a subgroup. However, we have the following Theorem Let G be a group and H G, K G. Then HK G HK = KH. Proof: ( ) First, note that for any subgroup L, we have L = L 1. Assume that HK G. Then HK = (HK) 1 = K 1 H 1 = KH. ( ) Assume that HK = KH. Then (HK) 2 = HKHK = HHKK = HK, (HK) 1 = K 1 H 1 = KH = HK. By Definition 1.1.2, we have HK G. Example Let G be the symmetry group of the regular tetrahedron having vertices 1, 2, 3, 4. The order of G is 12. In fact, it consists of the even permutations of four vertices 1, 2, 3, 4, and then G = A 4, the alternating group acting on the set {1, 2, 3, 4}. Let H = (123) and K = (12)(34). Since HK KH, HK is not a subgroup of G. In fact, HK = 6, but A 4 has no subgroup of order 6. (A 4 is also called the tetrahedron group. (Hint: if it had, then it would be normal; the intersection of this subgroup and the normal subgroup of order 4 would be a normal subgroup of order 2, impossible.) (The alternating group will be defined later in page 28.)
17 Groups and subgroups 7 Definition Let H G and a G. The subset ah (or Ha, resp.) is called a left coset (or right coset, resp.) of H. It is easy to see that ah = bh a 1 b H, while Ha = Hb ab 1 H. Proposition Let H G and let a, b G. Then (1) ah = bh ; (2) If ah bh then ah = bh. Now, the group G can be expressed as a disjoint union of the left cosets of H: G = a 1 H a 2 H a n H, where {a 1, a 2,..., a n }, the set of representatives of the left cosets of H in G, is called a transversal of the coset decomposition. The number n of distinct left cosets of H in G is called the index of H in G, denoted by G : H, which is not necessarily finite, The same conclusion is true for right cosets. So, G : H is also the number of right cosets of H in G. The following theorem is a basic property of subgroups of a finite group. Theorem (Lagrange) Let G be a group and let H G. Then G = H G : H. For a group G, the order of a subgroup of G is a divisor of G by Lagrange s Theorem In particular, the order o(a) of any element a is a divisor of G. The converse of Lagrange s Theorem does not hold, as shown in Example One might ask under what condition, the converse can hold. A partial answer will be mentioned later as Sylow Theorem, and the converse holds for any abelian or nilpotent group. (See Theorem1.2.9 and Corollary ) Also, the converse of Lagrange s Theorem holds for the symmetric group S 4 ; the verification will be left to the reader. For further information see Exercise
18 8 Basic concepts Theorem Let G be a group, and let H and K be two subgroups of G. Then HK = H K H K. Proof: Since HK is a disjoint union of right cosets of the form Hk, k K, and every right coset has H elements, it suffices to show that HK contains K : H K right cosets of H in G. It is easy to see that Noting that k 1 k 1 2 K, we have Hk 1 = Hk 2 k 1 k 1 2 H. Hk 1 = Hk 2 k 1 k2 1 H K (H K)k 1 = (H K)k 2. It follows that the number of right cosets of H in HK is equal to the index of H K in K, i.e., to K : H K. Proposition Let G be a group and let H G and K G. Then (1) H, K : H K : H K ; (2) G : H K = G : H H : H K G : H G : K ; (3) If G : H and G : K are relatively prime, then G : H K = G : H G : K and G = HK. Proof: (1) From the proof of Theorem 1.1.5, there are KH : H = K : H K right cosets of H in HK, here we slightly abuse to use the notation KH : H because KH is not necessarily a subgroup of G; in fact, we use KH : H to denote the number of cosets of H in KH. Since H, K HK, we have H, K : H HK : H = K : H K. (2) Since G : H K = G : K K : H K,
19 Groups and subgroups 9 and G : H H, K : H, we have G : H K : H K by (1), and hence G : H K G : H G : K. (3) By Lagrange s Theorem, G : H and G : K are divisors of G : H K. Since G : H and G : K are relatively prime, we have It follows from (2) that On the other hand, G : H G : K is a divisor of G : H K. G : H K = G : H G : K. G : H K = G : K K : H K = G : K HK : H, and hence G : H = HK : H, and G = HK. Let G be a group, and let a, g G. We write a g = g 1 ag. We call a g the conjugate of a by g. For a subgroup or a subset H of G, let H g = g 1 Hg, which is also called the conjugate of H by g. Two elements a and b (or two subgroups or subsets H and K) of G are called conjugate in G if there is an element g G such that a g = b (or H g = K). It is easy to see that the conjugacy relation between elements (or subgroups or subsets) is an equivalence relation. The corresponding equivalence classes C 1 = {1}, C 2,..., C k, are called the conjugacy classes of G, also we have G = C 1 C 2 C k (a disjoint union), and hence G = C 1 + C C k. The latter equation is called the class equation of G. The number of elements in C i, denoted by C i, is called the length of C i.
20 10 Basic concepts Definition Let G be a group and let H be a subset (or a subgroup) of G. (1) An element g G normalizes H if H g = H. We call the normalizer of H in G. N G (H) = {g G H g = H} (2) An element g G centralizes H if h g = h for any h H. We call C G (H) = {g G h g = h, h H} the centralizer of H in G. If H = G, then Z(G) := C G (G) is called the center of G. Note that for a subgroup H of G, C G (H) = G if and only if H Z(G). It is easy to see that, for any subset H, N G (H) and C G (H) are subgroups of G with N G (H) C G (H), and that if H G then H N G (H). When H = {a} is a oneelement set, we write N G (a) and C G (a) for N G (H) and C G (H), respectively. In this case, we have C G (a) = N G (a). Theorem Let G be a finite group. Then, (1) the conjugacy class C of G containing an element a has length C = G : C G (a), and hence C is a divisor of G. (2) The number of subgroups (or subsets) conjugate to a subgroup (or a subset) H equals the index G : N G (H) of the normalizer N G (H) in G, which is a divisor of G. Note that a conjugate of a subgroup H of G is a different concept from a conjugacy class of G. In fact, the conjugacy classes of G form a partition of a group G. However, the conjugates {H g g G} of H are neither disjoint each other nor a cover of G. (See Proposition ) Next, we generalize the concept of right or left cosets. Definition Let H and K be two (not necessarily distinct) subgroups of a group G. The subset HaK, a G, is called a double coset of G with respect to H and K, or simply an (H, K)double coset.
21 Groups and subgroups 11 Similar to cosets, we have Proposition For any a, b G, HaK HbK HaK = HbK. So G can be decomposed into a disjoint union of double cosets: G = Ha 1 K Ha 2 K Ha s K. The double coset HaK is a union of several right cosets of H, and also is a union of several left cosets of K. Theorem Let G be a group and let H and K be any two (not necessarily distinct) subgroups of G. Then the number of right cosets of H in double coset HaK is K / H a K, and the number of left cosets of K in HaK is H / H K a. Proof: We only prove the first statement. The number of right cosets of H in HaK is HaK / H. Since HaK = a 1 HaK = H a K, by Theorem 1.1.5, we have HaK = H a K / H a K = H K / H a K, from which the conclusion follows. Let G and H be two groups. A map α : G H is called a homomorphism of G into H if (ab) α = a α b α, a, b G. If a homomorphism α is surjective (or injective), then we call α an epimorphism (or monomorphism); if α is a bijection, then α is an isomorphism from G onto H. In this case, G and H are said to be isomorphic, denoted by G = H. A homomorphism and an isomorphism from G to itself is called an endomorphism and an automorphism of G, respectively. We use End(G) to denote the set of all endomorphisms of G, and Aut(G) the set of all automorphisms of G. For the composition of maps, End(G) is a monoid, and Aut(G) is a group, called the automorphism group of G.
22 12 Basic concepts For g G, the map σ(g) : G G defined by a σ(g) = a g, a G, is an automorphism of G, which is called the inner automorphism of G induced by g, denoted by Inn(g). The set Inn(G) of all inner automorphisms of G is a subgroup of Aut(G), called the inner automorphism group of G. The map σ : g Inn(g) is an epimorphism from G to Inn(G). Exercises Let G be a group, and let g G, o(g) = n. Then o(g m ) = n/(m, n) Let H G, g G. If o(g) = n and g m H, (n, m) = 1, then g H Let G be a group. If exp G = 2, then G is an abelian group Let H G, K G, and a, b G. If Ha = Kb, then H = K Let A, C be subgroups of G. If A C, then AB C = A(B C) for any subgroup B of G Let A, B, C be subgroups of G, and let A B. If A C = B C and AC = BC, then A = B Prove that a group having no nontrivial subgroups is cyclic of prime order Determine the conjugacy classes of A 4 and S (The number of conjugacy classes in G) The number of conjugacy classes in a finite group G is called the class number of G, denoted by k(g). E. Landau(1903) answered an open question posed by Frobenius by showing that for any positive integer k, there are only a finite number of finite groups G with k(g) = k. In relation to the class number k(g), let n(k) denote the maximum order of a group having k classes, and k(n) the minimum number of classes in a group of order n. Show that: (i) Prove that n(1) = 1, n(2) = 2, n(3) = 6, n(4) = 12 and n(5) = 60. (ii) For small k determine all finite groups with class number k. (iii) For a given n, find a lower bound for k(n); or equivalently, for a given k, find an upper bound for n(k). Landau proved n(k) k 2k, see Brauer (1963). For estimating k(n), see also Ayoub (1967), Sherman (1969) and Poland (1968) Show Aut(S n ) = S n except n = 6. Also, determine Aut(S 6 ). Furthermore, show that Aut(A n ) = Aut(S n ).
23 Normal subgroups and direct products Show that A n has a unique conjugacy class of subgroups isomorphic to A n 1 if n 6; and has exactly two such classes if n = Given a natural number n, it is one of the basic problems in group theory to determine all nonisomorphic groups of order n. With the aid of a computer, Besche, Eick and O Brien (2002) determined all groups of order The number of nonisomorphic groups of order n are 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, etc. For what n you can solve the same problem by hand now, or after studying the first three chapters of this book? For more information, visit The Small Groups library at hubesche/small.html. 1.2 Normal subgroups and direct products We study a subgroup of a group whose left cosets coincide with the right cosets. Definition A subgroup N of a group G is called normal, denoted by N G if N g N, g G. Obviously, every subgroup of an abelian group is normal. And any group G has at least two normal subgroups, that is G and 1, called the trivial normal subgroups. Proposition Let G be a group. Then the following statements are equivalent: (1) N G; (2) N g = N, g G (say, a selfconjugate subgroup); (3) N G (N) = G; (4) For each g N, we have Cl(g) N, where Cl(g) denotes the conjugacy class of G containing g. (5) Ng = gn, g G, that is, every left coset of N in G is also a right coset.
24 14 Basic concepts Definition A group is called simple if it has no nontrivial normal subgroups. Since each subgroup of an abelian group is normal, the only abelian simple groups are cyclic of prime order, (see Exercise 1.1.7). However, a classification of all nonabelian simple groups was one of long standing open problems in finite group theory. Although in Galois age some nonabelian simple groups, say A n for n 5, were known, all finite simple groups have been found in early 1980s. There are 18 infinite families of finite simple groups and 26 sporadic simple groups in totality. The classification theorem of finite simple groups asserts that every finite simple group is isomorphic to one of groups in the 18 families or one of the 26 sporadic groups. The proof of this theorem was scattered in about 500 papers. A readable proof is still in preparation. Very recently, M. Aschbacher and S. Smith successfully classified socalled finite quasithin simple groups in 2003, the classification of finite simple groups is finally completed. For more information we refer the reader to the wellorganized book, Gorenstein (1982). Also, we shall give more detailed description for finite simple groups in volume two. It was shown that for two subgroups H and K of a group G, HK is not necessary to be a subgroup, but it is a subgroup if at least one of H and K is normal, by Theorem The next proposition shows that it will be a normal subgroup if both H and K are normal. Proposition Let N 1, N 2,..., N s be normal subgroups of G. Then s i=1 N i and N 1, N 2,..., N s are also normal, and N 1, N 2,..., N s = N 1 N 2 N s. Definition Let G be a group and M G. We call M G = m g m M, g G the normal closure of M in G or the normal subgroup generated by M. It is easy to see that the normal closure M G consists of elements of the form l k=1 g 1 k m ε k k g k, g k G, m k M, ε k = ±1,
25 Normal subgroups and direct products 15 and hence M G is the smallest normal subgroup of G containing M. Assume that N G. Consider the set G = {Ng g G} of all (right) cosets of N in G. Define a multiplication in G as shown in Eq.(1.1), that is, the multiplication of subsets in G, namely, (Ng)(Nh) = N(gN)h = N(Ng)h = N 2 gh = Ngh. (1.2) Then G = {Ng g G} is a group with this multiplication. (The converse is also true. That is, if G is a group with the multiplication defined in Eq.(1.2), then N G.) It is called the factor group or quotient group of G modulo N, and denoted by G = G/N. For any positive integer n, nz is a normal subgroup of the integer group Z. The factor group Z n = Z/nZ = {0, 1,..., n 1} is called the (additive) group of integers modulo n. In this group, (nz)a (nz)b = (nz)c if and only a + b = c (mod n). Let α : G H be a group homomorphism. Then is called the kernel of α, and Ker α = {g G g α = 1} G α = {g α g G} is called the image of α. It is easy to see that Ker α G and G α H. Theorem (The Fundamental Theorem of Homomorphisms) (1) Let N G. Then the map ν : g Ng is a homomorphism of G onto G/N with Ker ν = N and G ν = G/N. Such a homomorphism ν is called the natural homomorphism from G onto G/N. (2) Let α : G H be a homomorphism. Then Ker α G and G α = G/Ker α. The next two theorems are the first and the second theorems of isomorphisms. Their proofs can be found in most textbooks of abstract algebra.
26 16 Basic concepts Theorem (The First Isomorphism Theorem) Let N G, M G and N M. Then M/N G/N and (G/N)/(M/N) = G/M. Theorem (The Second Isomorphism Theorem) G and N G. Then Let H (H N) H and HN/N = H/(H N). Given two groups G, H, their (outer) direct product is defined by G H = {(g, h) g G, h G}, where the multiplication is defined by (g, h)(g, h ) = (gg, hh ) for g, g G, h, h H. In the same way, one may define the (outer) direct product of n groups G 1,..., G n. Let G be the (outer) direct product of n groups G 1,..., G n : For i = 1, 2,..., n, let G = G 1 G n. H i = {(1,..., 1, g i, 1,..., 1) g i G i }, where g i is the ith component. Then H i = Gi, and the following hold: (1) H i G, i; (2) G = H 1, H 2,..., H n = H 1 H 2 H n ; (3) For i j, H i and H j are commutative elementwise, that is, for any h i H i and h j H j, h i h j = h j h i ; (4) H i (H 1 H i 1 H i+1 H n ) = 1, i; (5) Every element h of G can be uniquely expressed as a product of elements in H 1,..., H n : h = h 1 h n, h 1 H 1,..., h n H n.
27 Normal subgroups and direct products 17 Next, we define the inner direct product of subgroups. A group G is called the inner direct product of its subgroups H and K if G = HK and the map (h, k) hk is an isomorphism from the outer direct product H K onto G. For an inner direct product, we use the same notation as an outer direct product, that is G = H K. Similarly, a group G can be defined as the inner direct product of its n subgroups H 1,..., H n. Theorem Let G be a group and let H 1,..., H n be subgroups of G. Then the following are equivalent. (1) The map H 1 H n G defined by (h 1,..., h n ) h 1 h n is an isomorphism. (2) H i G for all i and each h G can be expressed uniquely as a product of elements of H 1,..., H n : h = h 1 h n, h 1 H 1,..., h n H n. (3) H i G for all i, G = H 1,..., H n, and for each i H i j i H j = 1. If any of the conditions (1)(3) holds, G is called the inner direct product of the subgroups H 1,..., H n. The condition (3) can be weakened to (3 ) H i (H 1 H i 1 ) = 1, i = 2,..., n. In other words, the equations (1), (2) and (3 ) are also a necessary and sufficient condition for G = H 1 H n. Also we have In fact, if G is the inner direct product of its n subgroup H 1,..., H n, it is isomorphic to their outer direct product. Proposition Let G be a group and M G, N G, M N = 1. Then for any m M and n N, we have mn = nm. Proof: Note that m 1 n 1 mn = (m 1 n 1 m)n = m 1 (n 1 mn). Since N G, m 1 n 1 m N; since M G, n 1 mn M. Therefore, m 1 n 1 mn M N. Since M N = 1, we have m 1 n 1 mn = 1, that is mn = nm.
28 18 Basic concepts Proposition Let M G, N G. Then G/(M N) (G/M) (G/N). (The notation H G means that H is isomorphic to a subgroup of G. ) Proof: Consider the map σ : G (G/M) (G/N) defined by g σ = (gm, gn), g G. Clearly, σ is a homomorphism from G to (G/M) (G/N) and Ker σ = M N. By the Fundamental Theorem of Homomorphisms, we have G/(M N) (G/M) (G/N). Let p be a prime number. An element a of a group G is called a pelement if its order o(a) is a power of p. A group G is called a p group if every element of G is a pelement. Later, it will be shown that a finite group is a pgroup if and only if its order is a power of p. (See Exercise ) More generally, for a set π of primes, a group G is called a πgroup if every prime divisor of G belongs to π. An element a G is called a πelement if a is πgroup. The πgroups will be studied later. Theorem (A classification of finite abelian groups) (1) A finite abelian pgroup G can be expressed as a direct product of cyclic subgroups: G = a 1 a s. The number s and the orders p e 1,..., p es of the direct factors (where one may assume that e 1 e s ) are uniquely determined by G. We call (p e 1,..., p es ) the type invariant of G, and call {a 1,..., a s } a basis for G. (2) A finite abelian group G can be expressed as a direct product of cyclic groups: G = a 1 a 2 a s, where o(a i ) o(a i+1 ), i = 1, 2,..., s 1.
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