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Extraspecial commutator-in-center subgroup is central factor

468 bytes added, 14:16, 24 September 2008
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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]])
'''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>gH/Z(H)</math> induces is in the identity map on the quotient center of <math>HG/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>gK</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.
===Textbook references===
* {{booklink-proved|Gorenstein}}, Page 195, Lemma 4.6, Section 5.4
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