Paper 3, Section II, $7 \mathrm{D}$

Let $G$ be a finite group of order $n$. Show that $G$ is isomorphic to a subgroup $H$ of $S_{n}$, the symmetric group of degree $n$. Furthermore show that this isomorphism can be chosen so that any nontrivial element of $H$ has no fixed points.

Suppose $n$ is even. Prove that $G$ contains an element of order 2 .

What does it mean for an element of $S_{m}$ to be odd? Suppose $H$ is a subgroup of $S_{m}$ for some $m$, and $H$ contains an odd element. Prove that precisely half of the elements of $H$ are odd.

Now suppose $n=4 k+2$ for some positive integer $k$. Prove that $G$ is not simple. [Hint: Consider the sign of an element of order 2.]

Can a nonabelian group of even order be simple?

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