Extraspecial p-group

A $𝑝$ -group $𝑃$ is called extraspecial iff its centre $Z (𝑃)$ has order $𝑝$ , and the quotient $𝑃 / 𝑍 (𝑃)$ is a nontrivial elementary abelian $𝑝$ -group, #m/def/group so if $| 𝑃 | = 𝑝 𝑛$

Z (𝑃) = [𝑃, 𝑃] ≅ ℤ + 𝑝 𝑃 / Z (𝑃) ≅ (ℤ + 𝑝) 𝑛 - 1

where $[𝑃, 𝑃]$ is the commutator subgroup of $𝑃$ . Equivalently, $𝑃$ is a central extension of the form

1 \to ℤ + 𝑝 ↪ 𝑃 ↠ 𝐸 \to 1

where $𝐸$ is an Elementary abelian group such that the associated commutator map $𝑐 0 : 𝐸 \times 𝐸 \to ℤ 𝑝$ is a nondegenerate $ℤ 𝑝$ -bilinear form.

Proof

Assume $𝑃$ is a $𝑝$ -group with $𝑍 (𝑃) ≅ ℤ + 𝑝$ and $𝑃 / 𝑍 (𝑃) ≅ (ℤ + 𝑝) 𝑛 - 1$ . Then by the Main theorem of abelianization, $[𝑃, 𝑃] ⊴ 𝑍 (𝑃)$ . Assume $𝑍 (𝑃) ⋬ [𝑃, 𝑃]$ . Then $[𝑃, 𝑃] = 1$ , which implies $𝑍 (𝑃) = 𝑃$ whence $𝑃 / 𝑍 (𝑃) = 1$ , a contradiction. Therefore $𝑍 (𝑃) = [𝑃, 𝑃]$ .

Now the commutator map $𝑐 0 : 𝐸 \times 𝐸 \to ℤ + 𝑝$ is nondegenerate iff for $𝑎 \in 𝐸$ ,

𝑐 0 (𝑎, 𝐸) = 0 ⟹ 𝑎 = 0

but

e 𝑐 0 (𝑎, 𝐸) = [𝑠 𝑎, 𝑠 𝐸] = [𝑠 𝑎 e ℤ + 𝑝, 𝑠 𝐸 e ℤ + 𝑝] = [𝜋 - 1 {𝑎}, 𝑃] = 1

implies $𝜋 - 1 {𝑎} \subseteq 𝑍 (𝑃)$ , in which case $𝑎 = 0$ , as required.

Special cases

2-central extension of an elementary abelian 2-group

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