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It has been suggested that this article 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.
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.
Properties [edit]
The trivial group is cyclic of order 1; as such it may be denoted Z1 or C1. If the group operation is called addition, the trivial group is usually denoted by 0. If the group operation is called multiplication then 1 can be a notation for the trivial group.
The trivial group serves as the zero object in the category of groups, meaning it is both an initial object and a terminal object.
See also [edit]
References [edit]
- Rowland, Todd and Weisstein, Eric W., "Trivial Group", MathWorld.
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