Frattini-in-center odd-order p-group implies p-power map is endomorphism

Statement
Suppose $$p$$ is an odd prime, and $$P$$ is a finite $$p$$-group (i.e., a group of prime power order) that is a Frattini-in-center group: the Frattini subgroup of $$P$$ is contained in its center. Then, the map $$x \mapsto x^p$$ is an endomorphism of $$P$$.

Note that this makes it a universal power endomorphism, i.e., an endomorphism described everywhere as raising to a certain power. The endomorphism is nontrivial only if $$P$$ does not itself have exponent $$p$$.

Examples
The smallest non-abelian examples for any odd prime $$p$$ are the two non-abelian groups of order $$p^3$$, namely unitriangular matrix group:UT(3,p) (GAP ID $$(p^3,3)$$) and semidirect product of cyclic group of prime-square order and cyclic group of prime order (GAP ID $$(p^3,4)$$). Of these two groups, the former has exponent $$p$$, so the $$p$$-power map is the trivial endomorphism. The latter has exponent $$p^2$$, so the $$p$$-power map is a nontrivial endomorphism.

In the case $$p = 3$$, these groups are unitriangular matrix group:UT(3,3) and semidirect product of Z9 and Z3 respectively. Both groups have order $$3^3 = 27$$.

Failure at the prime two

 * Square map is endomorphism iff abelian, combined with the fact that there exist non-Abelian 2-groups that are Frattini-in-center.

Facts with similar proofs

 * Omega-1 of odd-order class two p-group has prime exponent

Related facts about power maps

 * Cube map is endomorphism iff abelian (if order is not a multiple of 3)
 * Inverse map is automorphism iff abelian
 * Frattini-in-center odd-order p-group implies (p plus 1)-power map is automorphism

Facts used

 * 1) Frattini-in-center p-group implies derived subgroup is elementary abelian
 * 2) Formula for powers of product in group of class two

Proof
Given: An odd prime $$p$$. A finite $$p$$-group $$P$$, such that $$P/Z(P)$$ is elementary Abelian.

To prove: The map $$x \mapsto x^p$$ is an endomorphism of $$P$$. Specifically $$(xy)^p = x^py^p$$ for any $$x,y \in P$$.

Proof:

Textbook references

 * , Page 183-184, Lemma 3.9, Section 5.3 ($$p'$$-automorphisms of $$p$$-groups)