Definition:Trivial Group

Theorem
The trivial group is a group with only one element $e$.

It is (trivially) abelian and cyclic.

Proof
For a group $G = \left\{{e}\right\}$ to be a group, it follows that $e \circ e = e$.

Showing that $\left({G, \circ}\right)$ is in fact a group is straightforward:


 * $G$ is closed:


 * $\forall e \in G: e \circ e = e$


 * $e$ is the identity:


 * $\forall e \in G: e \circ e = e$


 * $\circ $ is associative:


 * $e \circ \left({e \circ e}\right) = e = \left({e \circ e}\right) \circ e$


 * Every element of $G$ (all one of them) has an inverse:

This follows from the fact that the identity is self-inverse, and the only element in $G$ is indeed the identity:


 * $e \circ e = e \implies e^{-1} = e$

Also see

 * Trivial Group is Normal