Special and extraspecial $p$-groups

Let $p$ be a prime and let $P$ be a finite $p$-group (A finite $p$-group has order $p^n$ for a prime $p$ and some $n\in \mathbb{N}$). Write $P^{\mathrm{\prime }}=[P,P]$ for its derived subgroup (Commutators $[g,h]=gh{g}^{-1}{h}^{-1}$ and the commutator subgroup $[G,G]$), $Z(P)$ for its centre (The center $Z(G)$ of a group) and $\mathrm{\Phi }(P)$ for its Frattini subgroup (The Frattini subgroup $\mathrm{\Phi }(G)$ as the intersection of the maximal subgroups of a finite group).

A finite $p$-group $P$ is special when $Z(P)=P^{\mathrm{\prime }}=\mathrm{\Phi }(P)$ is elementary abelian, and extraspecial when in addition $P$ is nonabelian and this common subgroup has order $p$. Elementary abelian $p$-groups are those of Elementary abelian $p$-groups, and the trivial group is one of them.

The nonabelian clause is stated rather than left implicit. It is not redundant for every formulation in the literature: the informal description "$P\mathrm{/}Z(P)$ is elementary abelian and $\mathrm{\mid }Z(P)\mathrm{\mid }=p$" is satisfied by the cyclic group of order $p$, whose centre is the whole group, so a definition phrased that way must exclude the abelian case by hand. With the clause $Z(P)=P^{\mathrm{\prime }}$ in force the exclusion is automatic, since an abelian group has trivial derived subgroup.

Two source conventions for extraspecial groups are in circulation and agree under the order-$p$ centre hypothesis. Craven asks that $Z(P)=P^{\mathrm{\prime }}=\mathrm{\Phi }(P)$ be elementary abelian and then that this common subgroup have order $p$; van Beek asks that $P^{\mathrm{\prime }}=Z(P)$ have order $p$ and that $P\mathrm{/}Z(P)$ have exponent $p$. Their equivalence for nonabelian $P$ is the content of Three equivalent descriptions of an extraspecial $p$-group.