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