Paper 3, Section II, D

Groups | Part IA, 2011

(a) Let GG be a finite group, and let gGg \in G. Define the order of gg and show it is finite. Show that if gg is conjugate to hh, then gg and hh have the same order.

(b) Show that every gSng \in S_{n} can be written as a product of disjoint cycles. For gSng \in S_{n}, describe the order of gg in terms of the cycle decomposition of gg.

(c) Define the alternating group AnA_{n}. What is the condition on the cycle decomposition of gSng \in S_{n} that characterises when gAng \in A_{n} ?

(d) Show that, for every n,An+2n, A_{n+2} has a subgroup isomorphic to SnS_{n}.

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