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Let G be the subgroup of Sy generated by the permutations α = a= (13)(27), B = (17) (25)(39). =

(e) Let K be a subgroup of G, such that H <K <G, where H is the subgroup from Part (b). Suppose that |K| # |G|, and |K| |H|. Determine the order of K, and give an example of a subgroup K matching all the above criteria. (f) Deduce that H is a normal subgroup of K, and that K is a normal subgroup of G. Use the second isomorphism theorem to prove that the quotient group KH/K is isomorphic to the trivial group.