# Nontrivial subgroup

From Groupprops

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 Nontrivial subgroup, all facts related to Nontrivial subgroup) |Survey articles about this | Survey articles about definitions built on this

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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]

## Definition

### Symbol-free definition

A subgroup of a group is termed nontrivial, if the subgroup is *not* the trivial group, i.e. it has more than one element.

### Definition with symbols

A subgroup of a group is termed *nontrivial* if is *not* the trivial group: the one-element group comprising the identity element.

Note that if the group itself is trivial, it cannot have any nontrivial subgroup.

### Opposite

The opposite of the property of being nontrivial is the property of being *trivial*, i.e. being the subgroup comprising only the identity element.