# Contracharacteristic subgroup

From Groupprops

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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]

## Definition

### Symbol-free definition

A subgroup of a group is termed a **contracharacteristic subgroup** if it satisfies the following equivalent conditions:

- Its characteristic closure in the whole group equals the whole group.
- It is not contained in any proper characteristic subgroup of the whole group.

## Relation with other properties

### Stronger properties

## Facts

- Every subgroup is contracharacteristic in its normal closure: In particular, every subgroup of a group is a contracharacteristic subgroup of a normal subgroup.