Extraspecial p-group

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

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

#state/tidy | #lang/en | #SemBr