Trivial group

In mathematics , a trivial group is a group consisting of one element. This item must be a group unit ; depending on the context, it is denoted 0 (if the group operation is addition), 1 (in the case when the group operation means multiplication) or e . A trivial group cannot be confused with an empty set , since the axioms of a group require a unit in it.

The term is often referred to as “ G does not contain non-trivial subgroups”, meaning that the only subgroups of G are the trivial subgroup and G itself.

The trivial group is cyclic , therefore, sometimes it is denoted as Z ₁ .

A trivial group is a null object in the category of groups , that is, it is both an initial and a terminal object.