# Power subgroup

From Groupprops

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

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 **power subgroup** if there exists an integer such that:

For a group of prime power order, the agemo subgroup are power subgroups if a product of powers is also a 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 | Potentially power 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 transitiveABOUT 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