# Direct power-closed characteristic subgroup

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

A subgroup $H$ of a group $G$ is termed a direct power-closed characteristic subgroup if for every cardinal $\alpha$, the subgroup $H^\alpha$ inside the direct power $G^\alpha$ (using the external direct product) is a characteristic subgroup of $G^\alpha$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
marginal subgroup follows from marginality is direct power-closed and marginal implies characteristic |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
finite direct power-closed characteristic subgroup |FULL LIST, MORE INFO