Definition:Trivial Subgroup

Theorem
For any group $\left({G, \circ}\right)$, the group whose underlying set is $\left\{{e}\right\}$, where $e$ is the identity of $\left({G, \circ}\right)$, is a subgroup of $\left({G, \circ}\right)$.

The group $\left({\left\{{e}\right\}, \circ}\right)$ is called the trivial subgroup of $\left({G, \circ}\right)$.

Proof
Using the One-step Subgroup Test:


 * 1) $e \in \left\{{e}\right\} \implies \left\{{e}\right\} \ne \varnothing$;
 * 2) $e \in \left\{{e}\right\} \implies e \circ e^{-1} = e \in \left\{{e}\right\}$.