# Centralizer-free subgroup

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

This article describes a property that arises as the conjunction of a subgroup property: self-centralizing subgroup with a group property (itself viewed as a subgroup property): centerless group

View a complete list of such conjunctions

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is said to be **centralizer-free** if its centralizer within the whole group is trivial.

### Definition with symbols

A subgroup of a group is said to be **centralizer-free** if is trivial.