# Self-centralizing direct factor

From Groupprops

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:

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