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) uses::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.