# 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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## 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.

## 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 $H \le K \le G$ 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 $H \le K \le G$ and $H$ is proper in $K$, $H$ is also proper in $G$.