In mathematics, a trivial group or zero group is a group that consists of a single element.
The single element of the trivial group is the identity element and so it is usually denoted as such:
If the group operation is denoted
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.
, the group that consists of only the identity element is a subgroup of
has no nontrivial proper subgroups" refers to the only subgroups of
The trivial group is cyclic of order
If the group operation is called addition, the trivial group is usually denoted by
If the group operation is called multiplication then
can be a notation for the trivial group.
Combining these leads to the trivial ring in which the addition and multiplication operations are identical and
The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.
The trivial group can be made a (bi-)ordered group by equipping it with the trivial non-strict order