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# <math>H</math> is an [[extraspecial group]]
# <math>[G,H] \le Z(H)</math>(i.e., <math>H</math> is a [[commutator-in-center subgroup]] of <math>G</math>)
Then <math>HC_G(H) = G</math>, i.e., <math>H</math> is a [[central factor]] of <math>G</math>.
==Definitions used==
===Extraspecial group===
===Central factor===
==Facts used==
* [[uses::Inner equals IA in extraspecial]]
'''Given''': A finite <math>p</math>-group <math>G</math>, a subgroup <math>H</math> such that <math>[G,H] \le Z(H)</math> and <math>H</math> is 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>g</math> induces the identity map on the quotient <math>H/Z(H)</math>. Since <math>H</math> is extraspecial, <math>Z(H) = H'</math>, so conjugation by <math>G</math> induces the identity map on the Abelianization <math>H/H'</math> of <math>H</math>. Hence, it is an [[IA-automorphism]] of <math>H</math>. Now, using the above fact, that for an extraspecial group, IA-automorphisms are the same as inner automorphisms, we conclude that conjugation by <math>g</math> induces an inner automorphism of <math>H</math>.
===Textbook references===
* {{booklink-proved|Gorenstein}}, Page 195, Lemma 4.6, Section 5.4
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