Extraspecial commutator-in-center subgroup is central factor
Contents
Statement
Statement with symbols
Let be a group. Suppose is a subgroup of satisfying the following two conditions:
- is an extraspecial group
- (i.e., is a commutator-in-center subgroup of )
Then , i.e., is a central factor of .
Definitions used
Extraspecial group
Central factor
Further information: central factor
A subgroup of a group is termed a central factor of if is normal in , and the following holds: consider the induced map , by conjugation by . Then, the image of under this map is precisely .
Equivalently, every inner automorphism of restricts to an inner automorphism of .
Facts used
- Extraspecial implies inner automorphism group is self-centralizing in automorphism group (Note: An equivalent formulation of this is IA equals inner in extraspecial)
Proof
Given: A group , a subgroup such that and is extraspecial.
To prove: is a central factor of
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 is equivalent to an inner automorphism as far as is concerned.
First, observe that since , is in the center of . Thus, is in the center of the subgroup of obtained by the action of on by conjugation. In particular, is in the centralizer of in . By fact (1), we see this forces that , completing the proof.
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 195, Lemma 4.6, Section 5.4