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:

$\log _{p}({\mbox{exponent of }}P/Z(P))\leq m\leq \log _{p}({\mbox{exponent of }}P)$

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.