Self-centralizing direct factor

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: self-centralizing subgroup and direct factor
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup of a group is termed a self-centralizing direct factor if it satisfies the following equivalent conditions:

1. It is both a self-centralizing subgroup (i.e., it contains its own centralizer) and a direct factor of the whole group.
2. It is a direct factor of the whole group such that the quotient group is a centerless group.
3. It is a direct factor of the whole group with a complement that is a centerless group.

Relation with other properties

Weaker properties

• Centerless-quotient normal subgroup
• Self-centralizing normal subgroup
• Self-centralizing subgroup