# Automorphisms of order 2 in finite groups with no elements of order $p^2$

Let $G$ be a finite group with no element of order $p^2$ for each prime $p$. Also suppose that $\vert G\vert\neq p$, for each prime $p$. Does there always exist an automorphism $\phi$ of order 2 such that for at least one subgroup of $G$ say $H$, we have $\phi(H)\neq H$?

• $\{1\}$ is a counter example.
– user583416
Apr 8 '17 at 7:54

A series of counterexamples is given by the cyclic groups $C_n$ where $n$ is a squarefree composite integer. Indeed $C_n$ can be written as $C_{p_1} \times C_{p_2} \times \ldots \times C_{p_n}$. Where the $C_{p_i}$ are minimal subgroups of $C_n$. Each automorphism of $C_n$ leaves $C_{p_i}$ invariant. I presume that these are the only counterexamples but I found no immediate proof.
Okay so look at $Z_2 \oplus Z_2$. The automorphisms of this group form $GL_2(F_2)$. Now look at $\phi$ which sends $(0,1)$ to $(1,0)$ and vice versa and keeps the others fixed. Take $H=(0,0),(1,0)$. Here $\phi^2$ is identity. This should do. There is no element of order $4$ in G.