Improper 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]


This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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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 H of a group G is termed improper if H = G.

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 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

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.