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===Statement with symbols===

Let <math>G</math> be ~~a finite <math>p</math>-group, i.e. ~~a [[group ~~of prime power order~~]]. Suppose <math>H</math> is a subgroup of <math>G</math> satisfying the following two conditions:

# <math>H</math> is an [[extraspecial group]]

==Facts used==

==Proof==

'''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 ~~Thus, conjugation by <math>~~H~~g</math> ~~is extraspecial~~, viewed as an automorphism of <math>~~Z(~~H~~) = H'~~</math>, ~~so conjugation by ~~commutes with all the inner automorphisms of <math>~~G~~H</math> ~~induces ~~. In particular, the ~~identity map on the Abelianization ~~automorphism induced by conjugation by <math>~~H/H'~~g</math> is in the centralizer of <math>\operatorname{Inn}(H)</math>~~. Hence, it is an [[IA-automorphism]] of ~~in <math>\operatorname{Aut}(H)</math>. ~~Now, using the above ~~By fact(1), ~~that for an extraspecial group, IA-automorphisms are ~~this forces the ~~same as inner automorphisms, we conclude that conjugation by <math>g</math> induces an inner ~~automorphism ~~of ~~to actually be in <math>\operatorname{Inn}(H)</math>.

==References==

===Textbook references===

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

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