Section 1.7. Group Actions 63 tst = si for i = 0, 1 or 2. Now i = 0, for tst = s0 = 1 implies s = 1. If i = 1, then s and t commute, and this gives st of order 6, as in Case 1 (which forces G to be cyclic, hence abelian, contrary to our present hypothesis). Therefore, tst = s2 = s−1. We construct an isomorphism G → S3. Let H = t , and consider the homo- morphism ϕ : G → S G/ t given by ϕ(g) : x t → gx t . By Theorem 1.96, ker ϕ ⊆ t , so that either ker ϕ = {1} (and ϕ is injective), or ker ϕ = t . Now G/ t = { t , s t , s2 t }, and, in two-rowed notation, ϕ(t) = t s t s2 t t t ts t ts2 t . If ϕ(t) is the identity permutation, then ts t = s t , so that s−1ts ∈ t = {1,t}, by Lemma 1.46. But now s−1ts = t (it cannot be 1) hence, ts = st, contradicting t and s not commuting. Therefore, t / ∈ ker ϕ, and ϕ: G → S G/ t ∼ = S3 is an injective homomorphism. Since both G and S3 have order 6, ϕ must be a bijection, and so G ∼ = S3. It is clear that I6 and S3 are not isomorphic, for I6 is abelian and S3 is not. • One consequence of this result is another proof that I6 ∼ = I2×I3 (Theorem 1.87). Classifying groups of order 8 is more diﬃcult, for we have not yet developed enough theory. Theorem 4.88 says there are only five nonisomorphic groups of order 8 three are abelian: I8, I4 × I2, and I2 × I2 × I2 two are nonabelian: D8 and Q. Order of Group Number of Groups 2 1 4 2 8 5 16 14 32 51 64 267 128 2, 328 256 56, 092 512 10, 494, 213 1024 49, 487, 365, 422 Table 4. Too many 2-groups. We can continue this discussion for larger orders, but things soon get out of hand, as Table 4 shows (Besche–Eick–O’Brien, The groups of order at most 2000, Electron. Res. Announc. AMS 7 (2001), 1–4). Making a telephone directory of groups is not the way to study them. Groups arose by abstracting the fundamental properties enjoyed by permu- tations. But there is an important feature of permutations that the axioms do

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