# Exponent of Self-similar finite $p$-groups

## TL;DR

This paper establishes that certain self-similar pro-$p$ groups with elements of bounded order are necessarily finite $p$-groups with bounded exponent, extending to power abelian $p$-groups.

## Contribution

It proves that self-similar pro-$p$ groups with nontrivial elements of bounded order are finite with bounded exponent, a new structural insight.

## Key findings

- Self-similar pro-$p$ groups with elements of order dividing $p^n$ are finite.
- Such groups have exponent at most $p^n$.
- The result applies to power abelian $p$-groups.

## Abstract

Let $p$ be a prime and $G$ a pro-$p$ group of finite rank that admits a faithful, self-similar action on the $p$-ary rooted tree. We prove that if the set $\{g\in G \ | \ g^{p^n}=1\}$ is a nontrivial subgroup for some $n$, then $G$ is a finite $p$-group with exponent at most $p^n$. This applies in particular to power abelian $p$-groups.