A general, and perhaps stupid, question: Can we have a group with one element? If so, what's an example?
EDIT: could a group with a single element and the identity function as the operation work?
Group (mathematics) - Wikipedia, the free encyclopedia:
"A trivial group is a group consisting of a single element. All such groups are isomorphic so one often speaks of the trivial group. The single element of the trivial group, variously labeled e, 1, or 0, is the identity element. The group operation is e + e = e."
See also Trivial Group -- from Wolfram MathWorld