8.3 Group Actions

1. a. If |G| = 20, there is a img G with o(a) = 5 by Cauchy's theorem. Thus |G : img a img|= 4 so there is a homomorphism θ : GS4 with ker θimg a img. Hence | img a img/ker θ| divides 24 so ker θ ≠ {1}. Thus ker θ =img a img because o(a) is prime, so imga img img G.
3. Assume pq. By Cauchy's theorem, let a

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