Trivial group

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In mathematics, a trivial group or zero 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 is the identity element and so it is usually denoted as such: ${\displaystyle 0,1,}$ or ${\displaystyle e}$ depending on the context. If the group operation is denoted ${\displaystyle \,\cdot \,}$ then it is defined by ${\displaystyle e\cdot e=e.}$

The similarly defined trivial monoid is also a group since its only element is its own inverse, and is hence the same as the trivial group.

The trivial group is distinct from the empty set, which has no elements, hence lacks an identity element, and so cannot be a group.