60 Chapter 1. Groups I ∗ 1.82. Generalize Theorem 1.87 as follows. Let G be a finite (additive) abelian group of order mn, where (m, n) = 1. Define Gm = {g ∈ G : order (g) | m} and Gn = {h ∈ G : order (h) | n}. (i) Prove that Gm and Gn are subgroups with Gm ∩ Gn = {0}. (ii) Prove that G = Gm + Gn = {g + h : g ∈ Gm and h ∈ Gn}. (iii) Prove that G ∼ = Gm × Gn. ∗ 1.83. Let G be a finite group, let p be prime, and let H be a normal subgroup of G. If both |H| and |G/H| are powers of p, prove that |G| is a power of p. 1.84. If H and K are normal subgroups of a group G with HK = G, prove that G/(H ∩ K) ∼ = (G/H) × (G/K). Hint. If ϕ: G → (G/H) × (G/K) is defined by x → (xH, xK), then ker ϕ = H ∩ K moreover, we have G = HK, so that a aH = HK = b bK. Definition. If H1, . . . , Hn are groups, then their direct product H1 × · · · × Hn is the set of all n-tuples (h1,.. . , hn), where hi ∈ Hi for all i, with coordinatewise multi- plication: (h1,... , hn)(h1,... , hn) = (h1h1,.. . , hnhn). ∗ 1.85. Let the prime factorization of an integer m be m = p11 e · · · pnn e . (i) Generalize Theorem 1.87 by proving that Im ∼ = Ipe1 1 × · · · × Ipnn e . (ii) Generalize Corollary 1.90 by proving that U(Im) ∼ = U(Ipe1 1 ) × · · · × U(Ipnn e ). ∗ 1.86. Define A, B ∈ GL(2, Q) by A = [ 0 −1 1 0 ] and B = 0 1 −1 1 . The quotient group M = A, B /N, where N = ±I , is called the modular group. (i) Show that a2 = 1 = b3, where a = AN and b = BN in M, and prove that ab has infinite order. (See Exercise 1.34 on page 27.) (ii) Prove that M ∼ = SL(2, Z)/N. Section 1.7. Group Actions Groups of permutations were our first examples of abstract groups the next result shows that abstract groups can be viewed as groups of permutations. Theorem 1.95 (Cayley). Every group G is isomorphic to a subgroup of the sym- metric group SG. In particular, if |G| = n, then G is isomorphic to a subgroup of Sn.

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