Power subgroup

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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]

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Definition

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

$H=\{g^{n}\mid g\in G\}$

For a group of prime power order, the $k^{th}$ agemo subgroup are power subgroups if a product of $(p^{k})^{th}$ powers is also a $(p^{k})^{th}$ power.

Relation with other properties

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Verbal subgroup				|FULL LIST, MORE INFO
1-endomorphism-invariant subgroup				|FULL LIST, MORE INFO
Fully invariant subgroup				|FULL LIST, MORE INFO
1-automorphism-invariant subgroup				|FULL LIST, MORE INFO
Quasiautomorphism-invariant subgroup				|FULL LIST, MORE INFO
Characteristic subgroup				|FULL LIST, MORE INFO
Normal subgroup				|FULL LIST, MORE INFO

Metaproperties

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity