Trivial group
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It has been suggested that this article or section be merged with trivial ring and trivial module to Zero object (algebra). (Discuss) Proposed since February 2012. |
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.
[edit] Properties
The trivial group is cyclic of order 1; as such it may be denoted Z1 or C1.
The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.
[edit] See also
[edit] References
Rowland, Todd and Weisstein, Eric W., "Trivial Group" from MathWorld.
