p-group

Extraspecial p-group

A p-group is called extraspecial iff its centre has order , and the quotient is a nontrivial elementary abelian -group, group so if

where is the commutator subgroup of . Equivalently, is a central extension of the form

where is an Elementary abelian group such that the associated commutator map is a ^nondegenerate -bilinear form.

Proof

Assume is a -group with and . Then by the Main theorem of abelianization, . Assume . Then , which implies whence , a contradiction. Therefore .

Now the commutator map is nondegenerate iff for ,

but

implies , in which case , as required.

Special cases

• 2 central extension of an elementary abelian 2-group

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