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 e ∗ e = e.
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 trivial group and being a subgroup of G is called the trivial subgroup of G.
The term, when referred to "G has no non-trivial subgroups" refers to the fact that all subgroups of G are the trivial group {e} and the group G itself.
Read more about Trivial Group: Properties
Famous quotes containing the words trivial and/or group:
“Individual science fiction stories may seem as trivial as ever to the blinder critics and philosophers of today—but the core of science fiction, its essence ... has become crucial to our salvation if we are to be saved at all.”
—Isaac Asimov (1920–1992)
“Laughing at someone else is an excellent way of learning how to laugh at oneself; and questioning what seem to be the absurd beliefs of another group is a good way of recognizing the potential absurdity of many of one’s own cherished beliefs.”
—Gore Vidal (b. 1925)