# Characterization of exponent semigroup of a finite p-group

## Statement=

Suppose $p$ is a prime number and $P$ is a finite p-group. Then, the exponent semigroup $\mathcal{E}(P)$ of $P$ is described as follows: there exists a nonnegative integer $m$ such that:

$\log_p(\mbox{exponent of } P/Z(P)) \le m \le \log_p(\mbox{exponent of } P)$

and:

$\mathcal{E}(P) = \left(\mbox{All multiples of } p^m\right) \cup \left(\mbox{All numbers that are of the form } 1 + kp^m, k \in \mathbb{Z} \right)$

## Facts used

1. nth power map is endomorphism implies every nth power and (n-1)th power commute

## Proof

The proof involves use of Fact (1) combined with the observation that if $n$ (respectively, $n-1$) is not a multiple of $p$, then every element of $P$ is a $n^{th}$ power (respectively, $(n-1)^{th}$ power).

The details are then simply possibility-chasing.