Root subgroup

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Definition

A subgroup $H$ of a group $G$ is termed a root subgroup if there exists an integer $n$ such that:

$H:=\{g\mid g^{n}=e\}$

When $G$ 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