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:
“My weakness has always been to prefer the large intention of an unskilful artist to the trivial intention of an accomplished one: in other words, I am more interested in the high ideas of a feeble executant than in the high execution of a feeble thinker.”
—Thomas Hardy (1840–1928)
“A little group of willful men, representing no opinion but their own, have rendered the great government of the United States helpless and contemptible.”
—Woodrow Wilson (1856–1924)