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)
“If the Russians have gone too far in subjecting the child and his peer group to conformity to a single set of values imposed by the adult society, perhaps we have reached the point of diminishing returns in allowing excessive autonomy and in failing to utilize the constructive potential of the peer group in developing social responsibility and consideration for others.”
—Urie Bronfenbrenner (b. 1917)