Extraspecial commutator-in-center subgroup is central factor
Statement with symbols
Let be a group. Suppose is a subgroup of satisfying the following two conditions:
Then , i.e., is a central factor of .
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 .
- Extraspecial implies inner automorphism group is self-centralizing in automorphism group (Note: An equivalent formulation of this is IA equals inner in extraspecial)
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.