# Improper subgroup

From Groupprops

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]

This article is about a basic definition in group theory. The article text may, however, contain advanced material.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Improper subgroup, all facts related to Improper subgroup) |Survey articles about this | Survey articles about definitions built on this

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## Definition

### Symbol-free definition

A subgroup of a group is termed **improper** if it equals the whole group.

### Definition with symbols

A subgroup of a group is termed **improper** if .

### Opposite

The negation of the subgroup property of being improper is the subgroup property of being proper.

## Metaproperties

### 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

The property of being improper is transitive: the improper subgroup of the improper subgroup is improper. In fact, it is a t.i. subgroup property.