# Changes

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

, 20:08, 13 July 2008
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# $H$ is an [[extraspecial group]]
# $[G,H] \le Z(H)$(i.e., $H$ is a [[commutator-in-center subgroup]] of $G$)
Then $HC_G(H) = G$, i.e., $H$ is a [[central factor]] of $G$.
==Definitions used==

===Extraspecial group===

===Central factor===

==Facts used==
* [[uses::Inner equals IA in extraspecial]]

==Proof==

'''Given''': A finite $p$-group $G$, a subgroup $H$ such that $[G,H] \le Z(H)$ and $H$ is extraspecial.

'''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 $g$ induces the identity map on the quotient $H/Z(H)$. Since $H$ is extraspecial, $Z(H) = H'$, so conjugation by $G$ induces the identity map on the Abelianization $H/H'$ of $H$. Hence, it is an [[IA-automorphism]] of $H$. Now, using the above fact, that for an extraspecial group, IA-automorphisms are the same as inner automorphisms, we conclude that conjugation by $g$ induces an inner automorphism of $H$.
==References==
===Textbook references===
* {{booklink-proved|Gorenstein}}, Page 195, Lemma 4.6, Section 5.4