# Kernel of a characteristic action on an abelian group

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]

## Contents

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

## Definition

A subgroup $H$ of a group $G$ is termed a kernel of a characteristic action on an abelian group if there exists an abelian group $V$ and a homomorphism $\alpha:G \to \operatorname{Aut}(V)$ with kernel $H$, such that $V$ is a characteristic subgroup of the semidirect product $V \rtimes G$.

## Relation with other properties

### Stronger properties

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