Trivial Subgroup

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

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 .

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.