Trivial group
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In mathematics, 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.
Every trivial group is abelian and cyclic; all these results being trivial, hence the name. The trivial group is often written as Z1 or just 0, 1, or e.
The subgroup of a group G consisting of just the identity element is called the trivial subgroup of G. It is, of course, a trivial group.
The trivial group should not be confused with the empty set (which has no elements and therefore, lacking an identity element, cannot be a group). The trivial group serves as the zero object in the category of groups. (The category of sets, on the other hand, has no zero objects; the empty set only serves as an initial object while any singleton set serves as a terminal object).
