Proper subgroup

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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VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 said to be proper if it does not equal the whole group, or equivalently, if as a subset, it is a proper subset of the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed proper if $H$ is not the whole of $G$ .

Opposite

The opposite of the property of being a proper subgroup is the property of being the improper subgroup, viz the whole group.

Relation with other properties

Related properties

Nontrivial subgroup is a subgroup that is not the trivial group

Metaproperties

Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.

Any subgroup of a proper subgroup is proper. That is, if $H\leq K\leq G$ and $K$ is proper in $G$ , so is $H$ .

Right-hereditariness

This subgroup property is right-hereditary: if a subgroup has the property in a group, it has the property in every bigger group. Hence, it is also a transitive subgroup property.

Any proper subgroup of a subgroup is proper. That is, if $H\leq K\leq G$ and $H$ is proper in $K$ , $H$ is also proper in $G$ .