IA equals inner in extraspecial
Further information: extraspecial group
Further information: IA-automorphism
An IA-automorphism of a group is an automorphism that induces the identity automorphism on the Abelianization of the group (the quotient by its commutator subgroup).
An equivalent fact
Given: An extraspecial -group . Let denote the group of IA-automorphisms of , and denote the group of inner automorphisms of
Proof: Note that (i.e., any inner automorphisms is IA). Also, (again, a standard fact). Since both groups are finite, it suffices to show that .
The first step in this is to show that elements of act as the identity. not only on the quotient , but also on . Thus, can be viewed as the stability group of the series . So, what we need to prove is that the cardinality of this stability group is at most .