74 Chapter 1. Groups I Exercises 1.87. If a and b are elements in a group G, prove that ab and ba have the same order. Hint. Use a conjugation. 1.88. Prove that if G is a finite group of odd order, then no x ∈ G, other than x = 1, is conjugate to its inverse. Hint. If x is conjugate to x−1, how many elements are in xG? 1.89. Prove that no two of the following groups of order 8 are isomorphic: I8 I4 × I2 I2 × I2 × I2 D8 Q. ∗ 1.90. Show that S4 has a subgroup isomorphic to D8. ∗ 1.91. Prove that S4/V ∼ = S3. Hint. Use Proposition 1.98. ∗ 1.92. (i) Prove that A4 ∼ = D12. Hint. Recall that A4 has no element of order 6. (ii) Prove that D12 ∼ = S3 × I2. Hint. Each element x ∈ D12 has a unique factorization of the form x = biaj , where b6 = 1 and a2 = 1. ∗ 1.93. (i) If G is a group, then a normal subgroup H G is called a maximal normal subgroup if there is no normal subgroup K of G with H K G. Prove that a normal subgroup H is a maximal normal subgroup of G if and only if G/H is a simple group. (ii) Prove that every finite abelian group G has a subgroup of prime index. Hint. Use Proposition 1.117. (iii) Prove that A6 has no subgroup of prime index. 1.94. (i) (Landau) Given a positive integer n and a positive rational q, prove that there are only finitely many n-tuples (i1,.. . , in) of positive integers with q = ∑n j=1 1/ij. (ii) Prove, for every positive integer n, that there are only finitely many finite groups having exactly n conjugacy classes. Hint. Use part (i) and the Class Equation. 1.95. Find NG(H) if G = S4 and H = (1 2 3) . ∗ 1.96. If H is a proper subgroup of a finite group G, prove that G is not the union of all the conjugates of H: that is, G = x∈G xHx−1. ∗ 1.97. (i) If H is a subgroup of G and x ∈ H, prove that CH (x) = H ∩ CG(x). (ii) If H is a subgroup of index 2 in a finite group G and x ∈ H, prove that either |xH | = |xG| or |xH | = 1 2 |xG|, where xH is the conjugacy class of x in H. Hint. Use the Second Isomorphism Theorem. (iii) Prove that there are two conjugacy classes of 5-cycles in A5, each of which has 12 elements.

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