The square map of an extraspecial $2$-group relative to a chosen generator of its centre

Let $P$ be an extraspecial $2$-group (Special and extraspecial $p$-groups) and fix the generator $z$ of its centre, so $Z(P)=\{1,z\}$ with $z^2=1$ (The center $Z(G)$ of a group, The order $\mathrm{\mid }G\mathrm{\mid }$ of a finite group and the order $\mathrm{ord}(g)$ of an element, with $\mathrm{ord}(g)=\mathrm{\infty }$ when no positive power of $g$ is the identity). Write $V=P\mathrm{/}Z(P)$ with its canonical ${\mathbb{F}}_2$-vector-space structure (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, An elementary abelian $p$-group has a canonical ${\mathbb{F}}_p$-vector-space structure, For every prime $p$, the two operations on $\mathbb{Z}\mathrm{/}p$ make it a field), and let $b_z$ be the commutator pairing (The commutator pairing of an extraspecial $p$-group relative to a chosen generator of its centre).

The square map of $P$ relative to $z$ is

$q_z:V\to {\mathbb{F}}_2,x^2=z^{q_z(\bar{x})}\mathrm{.}$

Why the exponent exists and is unique. The quotient $P\mathrm{/}Z(P)$ is elementary abelian (Three equivalent descriptions of an extraspecial $p$-group), so $x^2\in Z(P)=\{1,z\}$ for every $x\in P$; and $z^0=1\mathrm{\ne }z=z^1$, so exactly one class in ${\mathbb{F}}_2$ records which of the two values $x^2$ takes.

That the value depends only on the coset $\bar{x}$, and the identity relating it to the commutator pairing, are proved in The square map is well defined on the central quotient and satisfies $q(\bar{x}\bar{y})=q(\bar{x})+q(\bar{y})+b(\bar{x},\bar{y})$ ↗.

The map is not a homomorphism to ${\mathbb{F}}_2$: it fails additivity by exactly the value of the commutator pairing, and that failure is the whole content of the identity proved for it. Where the pairing vanishes the map is additive, and it is on such subspaces that the counting of elementary abelian subgroups is done.

The map is defined only at $p=2$. For odd $p$ the corresponding assignment $x\mapsto x^p$ also lands in the centre, but it is a homomorphism by the class-two power formula (For an odd prime $p$, the $p$-th power map is a homomorphism on a finite group whose derived subgroup is central of exponent dividing $p$) and carries no extra information beyond the type.