The maximal elementary abelian subgroups of the two extraspecial groups of order $2^{1+2n}$ have orders $2^{n+1}$ and $2^n$

Given: The hypotheses of the Statement.

For an extraspecial $2$-group $P$ of order $2^{1+2n}$ with $Z(P)=⟨z⟩$, the square map is the function $q$ determined by $x^2=z^{q(\bar{x})}$, with values in ${\mathbb{F}}_2$ (The square map of an extraspecial $2$-group relative to a chosen generator of its centre).

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 square map is well defined on the central quotient and satisfies $q(\bar{x}\bar{y})=q(\bar{x})+q(\bar{y})+b(\bar{x},\bar{y})$).

For a subgroup $\bar{A}$ of $P\mathrm{/}Z(P)$ one has $\mathrm{\mid }\bar{A}\mathrm{\mid }\mathrm{\mid }{\bar{A}}^{\perp }\mathrm{\mid }=\mathrm{\mid }P\mathrm{/}Z(P)\mathrm{\mid }$ (A subgroup of the central quotient and its orthogonal complement have orders multiplying to the order of the quotient).

For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$ up to isomorphism, with $2^{2n}+2^n$ and $2^{2n}-2^n$ solutions of $x^2=1$ (For each $n\ge 1$ there are exactly two extraspecial groups of order $2^{1+2n}$).

If $P_1$ and $P_2$ are extraspecial $2$-groups with $t_i$ solutions of $x^2=1$, then $P_1\circ P_2$ has $(t_1{t}_2+(\mathrm{\mid }{P}_1\mathrm{\mid }-t_1)(\mathrm{\mid }{P}_2\mathrm{\mid }-t_2))\mathrm{/}2$ such solutions (A product formula for the number of square roots of the identity in a central product of extraspecial $2$-groups).

Every maximal abelian subgroup of an extraspecial $p$-group of order $p^{1+2n}$ has order $p^{1+n}$ (In an extraspecial $p$-group of order $p^{1+2n}$ every maximal abelian subgroup has order $p^{1+n}$).

An extraspecial $p$-group has order $p^{1+2n}$ for some $n\ge 1$ (An extraspecial $p$-group has order $p^{1+2n}$ for some $n\ge 1$).

The commutator pairing is independent of the coset representatives, is ${\mathbb{F}}_p$-bilinear on $P\mathrm{/}Z(P)$, and is alternating (The commutator pairing is well defined on the central quotient, is bilinear over ${\mathbb{F}}_p$, and is alternating).

The commutator pairing of an extraspecial $p$-group has trivial radical (The commutator pairing of an extraspecial $p$-group has trivial radical).

For an extraspecial $p$-group $P$ with $Z(P)=⟨z⟩$, the commutator pairing is the map $b_z(\bar{x},\bar{y})\in \mathbb{Z}\mathrm{/}p$ determined by $[x,y]=z^{b_z(\bar{x},\bar{y})}$ (The commutator pairing of an extraspecial $p$-group relative to a chosen generator of its centre).

An elementary abelian $p$-group is a finite abelian $p$-group in which every nonidentity element has order $p$; the trivial group is permitted (,, ). (Elementary abelian $p$-groups).

The set $S$ is independent when ${\prod }_{s\in S}{s}^{a_s}=e$ with finite support forces every $a_s=0$. A basis of an elementary abelian $p$-group is an independent spanning subset for its canonical ${\mathbb{F}}_p$-linear structure. (${\mathbb{F}}_p$-spanning sets, independence, and bases in an elementary abelian $p$-group).

Every finite elementary abelian $p$-group has a basis; every independent subset extends to a basis, every spanning subset contains a basis, and all bases have the same finite size. (Finite elementary abelian $p$-groups have bases, basis extension, and a well-defined dimension).

$\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }\mathrm{.}$ (Lagrange's theorem: $\mathrm{\mid }G\mathrm{\mid }=[G:H]\mathrm{\mid }H\mathrm{\mid }$ for every subgroup $H$ of a finite group $G$).

$⟨S⟩;:=;\bigcap \{H;:;H\le G\text{ and }S\subseteq H\}\mathrm{.}$ (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups).

$Z(G):=\{z\in G:zg=gz\text{ for every }g\in G\}\mathrm{.}$ (The center $Z(G)$ of a group).

An extraspecial $2$-group of order $2^{1+2n}$ is of plus type when it has $2^{2n}+2^n$ solutions of $x^2=1$ and of minus type when it has $2^{2n}-2^n$ such solutions; for odd $p$, plus and minus mean exponent $p$ and $p^2$ respectively (Plus and minus type of an extraspecial $p$-group).

A central product of extraspecial $p$-groups identified along their centres is extraspecial (A central product of extraspecial $p$-groups identified along their centres is extraspecial).

Every extraspecial $p$-group is an internal central product of nonabelian subgroups of order $p^3$ pairwise intersecting in its centre (Every extraspecial $p$-group is an internal central product of nonabelian subgroups of order $p^3$).