What is "trivial group"

n. (context group theory English) the unique (l en group) (up to (l en isomorphism)) consisting of a single element (which is the (l en identity element))

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 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 ∗ then it is defined by .

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 should not be confused with the empty set (which has no elements, and lacking an identity element, cannot be a group).

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 trivial subgroup 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.