Trivial Subgroup - Imported Knowlpedia Content

Let $G$ be a group with identity element $e$ . The trivial subgroup of $G$ is the subgroup $\{e\}$ .

Examples

In $(\mathbb{Z},+)$ , the trivial subgroup is $\{0\}$ .
In $(\mathbb{R}^{\times},\times)$ , the trivial subgroup is $\{1\}$ .
In any permutation group, the trivial subgroup consists of the identity permutation only.

Remarks

The trivial subgroup is the smallest subgroup of $G$ (by inclusion). It appears as an endpoint in definitions such as that of a simple group.