Section 1.5. Homomorphisms 43 Example 1.66. If G is a group, then an automorphism25 of G is an isomorphism f : G → G. For example, every conjugation γg is an automorphism of G (it is called an inner automorphism) its inverse is conjugation by g−1. An automorphism is called outer if it is not inner. The set Aut(G) of all the automorphisms of G is itself a group under composition, called the automorphism group, and the set of all conjugations, Inn(G) = {γg : g ∈ G}, is a subgroup of Aut(G) (see Proposition 1.105). Example 1.67. The four-group V = (1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3) is a normal subgroup of S4. By Theorem 1.9, every conjugate of a product of two trans- positions is another such Table 1 on page 10 shows that only three permutations in S4 have this cycle structure, and so V is a normal subgroup of S4. Proposition 1.68. Let H be a subgroup of index 2 in a group G. (i) g2 ∈ H for every g ∈ G. (ii) H is a normal subgroup of G. Proof. (i) Since H has index 2, there are exactly two cosets, namely, H and aH, where a / ∈ H. Thus, G is the disjoint union G = H ∪ aH. Take g ∈ G with g / ∈ H, so that g = ah for some h ∈ H. If g2 / ∈ H, then g2 = ah , where h ∈ H. Hence, g = g−1g2 = h−1a−1ah = h−1h ∈ H, and this is a contradiction. (ii) 26 It suﬃces to prove that if h ∈ H, then the conjugate ghg−1 ∈ H for every g ∈ G. If g ∈ H, then ghg−1 ∈ H, because H is a subgroup. If g / ∈ H, then g = ah0, where h0 ∈ H (for G = H ∪ aH). If ghg−1 ∈ H, we are done. Otherwise, ghg−1 = ah1 for some h1 ∈ H. But ah1 = ghg−1 = ah0hh0 −1a−1. Cancel a to obtain h1 = h0hh0 −1 a−1, contradicting a / ∈ H. • Definition. The group of quaternions27 is the group Q of order 8 consisting of the following matrices in GL(2, C): Q = { I, A, A2,A3,B,BA,BA2,BA3 }, where I is the identity matrix, A = 0 1 −1 0 , and B = [ 0 i i 0 ]. The element A ∈ Q has order 4, so that A is a subgroup of order 4 and, hence, of index 2 the other coset is B A = {B, BA, BA2,BA3 }. Note that B2 = A2 and BAB−1 = A−1. 25The word automorphism is made up of two Greek roots: auto, meaning “self,” and morph, meaning “shape” or “form.” Just as an isomorphism carries one group onto a faithful replica, an automorphism carries a group onto itself. 26Another proof of this is given in Exercise 1.57 on page 45. 27Hamilton invented an R-algebra (a vector space over R which is also a ring) that he called quaternions, for it was four-dimensional. The group of quaternions consists of eight special ele- ments in that system see Exercise 1.68 on page 47.

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