p-group

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, group so if |𝑃| =𝑝𝑛

Z⁡(𝑃)=[𝑃,𝑃]≅ℤ+𝑝𝑃/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→ℤ+𝑝↪𝑃↠𝐸→1

where 𝐸 is an Elementary abelian group such that the associated commutator map 𝑐0 :𝐸 ×𝐸 →ℤ𝑝 is a ^nondegenerate ℤ𝑝-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 𝑐0 :𝐸 ×𝐸 →ℤ+𝑝 is nondegenerate iff for 𝑎 ∈𝐸,

𝑐0(𝑎,𝐸)=0⟹𝑎=0

but

e𝑐0(𝑎,𝐸)=[𝑠𝑎,𝑠𝐸]=[𝑠𝑎eℤ+𝑝,𝑠𝐸eℤ+𝑝]=[𝜋−1{𝑎},𝑃]=1

implies 𝜋−1{𝑎} ⊆𝑍(𝑃), in which case 𝑎 =0, as required.

Special cases

• 2 central extension of an elementary abelian 2-group

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