Cocentral implies central factor
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., cocentral subgroup) must also satisfy the second subgroup property (i.e., central factor)
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A subgroup of a group is a cocentral subgroup if where is the center of .
Further information: central factor
A subgroup of a group is a central factor if where is the centralizer of in .
- Cocentrality is upward-closed
- Cocentral implies right-quotient-transitively central factor
- Cocentral implies join-transitively central factor
Given: A group , a subgroup of such that where is the center of .
To prove: where is the centralizer of in .
Proof: We have , because any element of the center centralizes every element of . Thus, . Since , we obtain .