I want to form a group of order one, is it possible?

Question

Can a set with one element be a group, or does a group need to have at least two elements?

Answer 1

It is possible to have an element with just one element. Take $G=\{e\}$ with the group law defined as $e\circ e=e$. Then it is easy to verify that:

1. The group law is associative: $(e\circ e)\circ e=e\circ(e\circ e)=e$
2. There is an identity $e$ such that $e\circ g=g = g \circ e$ for all $g\in G$.
3. Every element of $G$ has an inverse, since $e$ is its own inverse.