Section 1.6. Quotient Groups 59 Exercises ∗ 1.72. Recall that U(Im) = {[r] ∈ Im : (r, m) = 1} is a multiplicative group. Prove that U(I9) ∼ = I6 and U(I15) ∼ = I4 × I2. (Theorem 4.24 says, for every finite abelian group G, that there exists an integer m with G isomorphic to a subgroup of U(Im).) 1.73. (i) Let H and K be groups. Without using the First Isomorphism Theorem, prove that H∗ = {(h, 1) : h ∈ H} and K∗ = {(1,k) : k ∈ K} are normal subgroups of H × K with H ∼ = H∗ and K ∼ = K∗, and f : H → (H × K)/K∗, defined by f(h) = (h, 1)K∗, is an isomorphism. (ii) Use Proposition 1.85 to prove that K∗ (H × K) and (H × K)/K∗ ∼ = H. Hint. Consider the function f : H × K → H defined by f : (h, k) → h. 1.74. (i) Prove that every subgroup of Q × I2 is normal (see the discussion on page 44). (ii) Prove that there exists a nonnormal subgroup of G = Q × I4. Conclude that G is not hamiltonian. 1.75. (i) Prove that Aut(V) ∼ = S3 and that Aut(S3) ∼ = S3. Conclude that nonisomorphic groups can have isomorphic automorphism groups. (ii) Prove that Aut(Z) ∼ = I2. Conclude that an infinite group can have a finite auto- morphism group. 1.76. (i) If G is a group for which Aut(G) = {1}, prove that g2 = 1 for all g ∈ G. (ii) If G is a group, prove that Aut(G) = {1} if and only if |G| ≤ 2. Hint. By (i), G can be viewed as a vector space over F2. You may use Corol- lary 5.50, which states that every GL(V ) = {1} for every infinite-dimensional vector space V . ∗ 1.77. Prove that if G is a group for which G/Z(G) is cyclic, where Z(G) denotes the center of G, then G is abelian that is, G/Z(G) = {1}. Hint. If G/Z(G) is cyclic, prove that a generator gives an element outside of Z(G) which commutes with each element of G. ∗ 1.78. (i) Prove that Q/Z(Q) ∼ = V, where Q is the group of quaternions and V is the four-group conclude that the quotient of a group by its center can be abelian. (ii) Prove that Q has no subgroup isomorphic to V. Conclude that the quotient Q/Z(Q) is not isomorphic to a subgroup of Q. 1.79. Let G be a finite group with K G. If (|K|, [G : K]) = 1, prove that K is the unique subgroup of G having order |K|. Hint. If H ⊆ G and |H| = |K|, what happens to elements of H in G/K? ∗ 1.80. If H and K are subgroups of a group G, prove that HK is a subgroup of G if and only if HK = KH. Hint. Use the fact that H ⊆ HK and K ⊆ HK. ∗ 1.81. Let G be a group and regard G × G as the direct product of G with itself. If the multiplication μ: G × G → G is a group homomorphism, prove that G must be abelian.

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