# Burnside's theorem on coprime automorphisms and Frattini subgroup

## Statement

Suppose $P$ is a finite $p$-group, i.e., a group of prime power order. Suppose $\varphi$ is an automorphism of $P$ whose order (as an element of the automorphism group of $P$) is relatively prime to $P$. Then, if $\varphi$ induces the identity on $P/\Phi(P)$, $\varphi$ is the identity automorphism of $P$. Here $\Phi(P)$ is the Frattini subgroup of $P$.

Equivalently the kernel of the map: $\operatorname{Aut}(P) \to \operatorname{Aut}(P/\Phi(P))$

is a $p$-group. We also say that the Frattini subgroup of a $p$-group is a quotient-coprime automorphism-faithful subgroup.

## Facts used

1. Frattini subgroup of finite group is quotient-coprime automorphism-faithful

## Proof

Given: A finite $p$-group $P$, an automorphism $\varphi$ of $P$ whose order is relatively prime to $p$. Further $\varphi$ induces the identity automorphism on $P/\Phi(P)$.

To prove: $\varphi$ is the identity automorphism.

Proof: The proof follows directly from fact (1), which is actually the general formulation for finite groups.

(NOTE: We can use the Burnside's basis theorem instead; instead of picking a coset representative for every coset, we pick a Burnside basis, and then find a coset representative for each coset in the Burnside basis, that is fixed under $\sigma$.)