Exercise 23, § 6

Let $\sigma$ be an automorphism of a group G. a) Now prove the equivalence of the following conditions: (i) $\sigma(x) = x$ entails $x = e$. (ii) The mapping $x \mapsto x^{-1}\sigma(x)$ is injective. When these conditions are satisfied, one says (by abuse of language) that $\sigma$ is without fixed point. b) Suppose G is finite and $\sigma$ is without fixed point. The mapping $x \mapsto x^{-1}\sigma(x)$ is then bijective. If H is a normal subgroup of G stable by $\sigma$, now prove that the automorphism of G/H defined by $\sigma$ is without fixed point. c) The hypotheses being those of b), let $p$ be a prime number. Now prove that there exists a $p$-Sylow subgroup P of G stable by $\sigma$. (If $P_0$ is a $p$-Sylow 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 $P = xP_0x^{-1}$.) Now prove that such a subgroup is unique, and contains every $p$-subgroup of G stable by $\sigma$. d) Suppose in addition that $\sigma$ is of order 2. Now prove 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.