Characterization of exponent semigroup of a finite p-group

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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.