Exercise 29, § 5

Let $X$ be an infinite set. Let $G$ be the subgroup of $\mathfrak{S}_X$ consisting of the permutations $\sigma$ which enjoy the following property: there exists a finite subset $Y_\sigma$ of $X$ such that $\sigma x = x$ if $x \in X - Y_\sigma$ and the restriction of $\sigma$ to $Y_\sigma$ is even.

(a) Show that G is a non-commutative simple group. Show that every subset of G which generates G is equipotent to X.

(b) Show that every finite group is isomorphic to a subgroup of G (use the fact that $\mathfrak{S}_n$ is isomorphic to a subgroup of $\mathfrak{A}_{n+2}$).