# Kernel of a characteristic action on an abelian group

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]

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

## Definition

A subgroup of a group is termed a **kernel of a characteristic action on an abelian group** if there exists an abelian group and a homomorphism with kernel , such that is a characteristic subgroup of the semidirect product .

## Relation with other properties

### Stronger properties

- Normal subgroup of finite group
- Normal subgroup of a group having no nontrivial abelian normal -subgroup for some prime .