Nontrivial subgroup

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 nontrivial, if the subgroup is not the trivial group, i.e. it has more than one element.

Definition with symbols

A subgroup $H$ of a group $G$ is termed nontrivial if $H$ 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.

Relation with other properties

Stronger properties

• Nontrivial normal subgroup