Paper 3, Section II, D

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

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

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

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

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