Trivial group

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: $$0, 1,$$ or $$e$$ depending on the context. If the group operation is denoted $$\, \cdot \,$$ then it is defined by $$e \cdot e = e.$$

The similarly defined  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.

Definitions
Given any group $$G,$$ the group consisting of only the identity element is a subgroup of $$G,$$ and, being the trivial group, is called the  of $$G.$$

The term, when referred to "$$G$$ has no nontrivial proper subgroups" refers to the only subgroups of $$G$$ being the trivial group $$\{ e \}$$ and the group $$G$$ itself.

Properties
The trivial group is cyclic of order $$1$$; as such it may be denoted $$\mathrm{Z}_1$$ or $$\mathrm{C}_1.$$ If the group operation is called addition, the trivial group is usually denoted by $$0.$$ If the group operation is called multiplication then 1 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 $$0 = 1.$$

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 $$\,\leq.$$