# Root 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

A subgroup of a group is termed a **root subgroup** if there exists an integer such that:

When is a finite group, a root subgroup is the same as a variety-containing subgroup of finite group.

## Relation with other properties

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Variety-containing subgroup | ||||

1-endomorphism-invariant subgroup | |FULL LIST, MORE INFO | |||

Existentially bound-word subgroup | |FULL LIST, MORE INFO | |||

Quasiendomorphism-invariant subgroup | |FULL LIST, MORE INFO | |||

Fully invariant subgroup | |FULL LIST, MORE INFO | |||

Characteristic subgroup | |FULL LIST, MORE INFO | |||

Normal subgroup | |FULL LIST, MORE INFO |