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.

Relation with other properties

Stronger properties

• Nontrivial normal subgroup