# Changes

## Extraspecial commutator-in-center subgroup is central factor

, 14:16, 24 September 2008
no edit summary
===Central factor===

{{further|[[central factor]]}}

A subgroup $H$ of a group $G$ is termed a central factor of $G$ if $H$ is normal in $G$, and the following holds: consider the induced map $G \to \operatorname{Aut}(H)$, by conjugation by $G$. Then, the image of $G$ under this map is precisely $\operatorname{Inn}(H)$.

Equivalently, every inner automorphism of $G$ restricts to an inner automorphism of $H$.
==Facts used==
# [[uses::Extraspecial implies inner automorphism group is self-centralizing in automorphism group]] (Note: An equivalent formulation of this is [[IA equals inner in extraspecial]])

==Proof==
'''To prove''': $H$ is a [[central factor]] of $G$
'''Proof''': We use the definition of central factor in terms of inner automorphisms. In other words, we strive to show that conjugation by any element of $G$ is equivalent to an inner automorphism as far as $H$ is concerned. So, pick a $g \in G$.
First, observe that since $[G,H] \le Z(H)$, conjugation by $gH/Z(H)$ induces is in the identity map on the quotient center of $HG/Z(H)$. Thus, conjugation by $g\operatorname{Inn}(H)$ is in the center of the subgroup $K$, viewed as an automorphism of $\operatorname{Aut}(H)$, commutes with all obtained by the inner automorphisms action of $G$ on $H$by conjugation. In particular, the automorphism induced by conjugation by $gK$ is in the centralizer of $\operatorname{Inn}(H)$ in $\operatorname{Aut}(H)$. By fact (1), we see this forces the automorphism to actually be in that $K = \operatorname{Inn}(H)$, completing the proof.
==References==
===Textbook references===
* {{booklink-proved|Gorenstein}}, Page 195, Lemma 4.6, Section 5.4