Exercise 23, § 6

Let $\sigma$ be an automorphism of a group $G$. (a) Show the equivalence of the following conditions: (i) $\sigma(x) = x$ implies $x = e$. (ii) The mapping $x \mapsto x^{-1}\sigma(x)$ is injective. When these conditions are satisfied, $\sigma$ is said (by an abuse of language) to be without fixed point. (b) Suppose $G$ is finite and $\sigma$ without fixed point. The mapping $x \mapsto x^{-1}\sigma(x)$ is then bijective. If $H$ is a normal subgroup of $G$ stable under $\sigma$, show that the automorphism of $G/H$ defined by $\sigma$ is without fixed point. (c) Under the hypotheses of (b), let $p$ be a prime number. Show that there exists a Sylow $p$-subgroup $P$ of $G$ which is stable under $\sigma$. (If $P_0$ is a Sylow $p$-subgroup, there exists $y \in G$ such that $\sigma(P_0) = yP_0y^{-1}$; write $y^{-1}$ in the form $x^{-1}\sigma(x)$ and take $P = xP_0x^{-1}$.) Show that such a subgroup is unique and contains every $p$-subgroup of $G$ which is stable under $\sigma$. (d) Suppose further that $\sigma$ is of order 2. Show that $\sigma(g) = g^{-1}$ for all $g \in G$ (write $g$ in the form $x^{-1}\sigma(x)$ with $x \in G$). Deduce that $G$ is commutative and of odd order.