# Proper 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 Proper subgroup, all facts related to Proper 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 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 of a group is termed **proper** if is not the whole of .

### 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 and is *proper* in , so is .

### 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 and is *proper* in , is also proper in .