44 Chapter 1. Groups I Proposition 1.69. The group Q of quaternions is not abelian, yet every subgroup of Q is normal. Proof. By Exercise 1.67 on page 46, Q is a nonabelian group of order 8 having exactly one subgroup of order 2, namely, the center Z(Q) = −I , which is nor- mal. Lagrange’s Theorem says that the only possible orders of subgroups are 1, 2, 4, or 8. Clearly, the subgroups {I} and Q itself are normal subgroups and, by Proposition 1.68(ii), any subgroup of order 4 is normal, for it has index 2. A nonabelian finite group is called hamiltonian if every subgroup is normal. The group Q of quaternions is essentially the only hamiltonian group, for every hamiltonian group has the form Q×A×B, where a2 = 1 for all a A (Exercise 1.35 on page 27 says that A is necessarily abelian) and B is an abelian group of odd order (direct products will be introduced in the next section) (Robinson, A Course in the Theory of Groups, p. 143). Lagrange’s Theorem states that the order of a subgroup of a finite group G must be a divisor of |G|. This suggests the question, given a divisor d of |G|, whether G must contain a subgroup of order d. The next result shows that there need not be such a subgroup. Proposition 1.70. The alternating group A4 is a group of order 12 having no subgroup of order 6. Proof. First, |A4| = 12, by Example 1.48(i). If A4 contains a subgroup H of order 6, then H has index 2, and so α2 H for every α A4, by Proposition 1.68(i). But if α is a 3-cycle, then α has order 3, so that α = α4 = (α2)2. Thus, H contains every 3-cycle. This is a contradiction, for there are eight 3-cycles in A4. Exercises 1.46. Show that if there is a bijection f : X Y (that is, if X and Y have the same number of elements), then there is an isomorphism ϕ: SX SY . Hint. If α SX , define ϕ(α) = fαf −1. In particular, show that if |X| = 3, then ϕ takes a cycle involving symbols 1, 2, 3 into a cycle involving a, b, c, as in Example 1.57. 1.47. (i) Show that the composite of homomorphisms is itself a homomorphism. (ii) Show that the inverse of an isomorphism is an isomorphism. (iii) Show that two groups that are isomorphic to a third group are isomorphic to each other. (iv) Prove that isomorphism is an equivalence relation on any set of groups. 1.48. Prove that a group G is abelian if and only if the function f : G G, given by f(a) = a−1, is a homomorphism. 1.49. This exercise gives some invariants of a group G. Let f : G H be an isomorphism.
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