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