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:
“In our most trivial walks, we are constantly, though unconsciously, steering like pilots by certain well-known beacons and headlands, and if we go beyond our usual course we still carry in our minds the bearing of some neighboring cape; and not till we are completely lost, or turned round,—for a man needs only to be turned round once with his eyes shut in this world to be lost,—do we appreciate the vastness and strangeness of nature.”
—Henry David Thoreau (1817–1862)
“Stripped of ethical rationalizations and philosophical pretensions, a crime is anything that a group in power chooses to prohibit.”
—Freda Adler (b. 1934)