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===Central factor===

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

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

Equivalently, every inner automorphism of <math>G</math> restricts to an inner automorphism of <math>H</math>.

==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''': <math>H</math> is a [[central factor]] of <math>G</math>

'''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 <math>G</math> is equivalent to an inner automorphism as far as <math>H</math> is concerned~~. So, pick a <math>g \in G</math>~~.

First, observe that since <math>[G,H] \le Z(H)</math>, ~~conjugation by ~~<math>~~g~~H/Z(H)</math> ~~induces ~~is in the ~~identity map on the quotient ~~center of <math>~~H~~G/Z(H)</math>. Thus, ~~conjugation by ~~<math>~~g~~\operatorname{Inn}(H)</math> is in the center of the subgroup <math>K</math>~~, viewed as an automorphism ~~of <math>\operatorname{Aut}(H)</math>~~, commutes with all ~~obtained by the ~~inner automorphisms ~~action of <math>G</math> on <math>H</math>by conjugation. In particular, ~~the automorphism induced by conjugation by ~~<math>~~g~~K</math> is in the centralizer of <math>\operatorname{Inn}(H)</math> in <math>\operatorname{Aut}(H)</math>. By fact (1), we see this forces ~~the automorphism to actually be in ~~that <math>K = \operatorname{Inn}(H)</math>, completing the proof.

==References==

===Textbook references===

* {{booklink-proved|Gorenstein}}, Page 195, Lemma 4.6, Section 5.4

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