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trivial group

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Definition

The trivial group is the group whose underlying set is the singleton, hence whose only element is the neutral element.

In the context of nonabelian groups the trivial group is usually denoted 11, while in the context of abelian groups it is usually denoted 00.

The trivial group is a zero object (both initial and terminal) of Grp.

Examples

The trivial group is a subgroup of any other group, and the corresponding inclusion 1G1 \hookrightarrow G is the unique such group homomorpism.

The quotient group of any group GG by itself is the trivial group: G/G=1G/G = 1, and the quotient projection GG/G=1G \to G/G =1 is the unique such group homomorphism.

It can be nontrivial to decide from a group presentation whether a group so presented is trivial, and in fact the general problem is undecidable. See also combinatorial group theory and word problem.

Properties

The trivial group is an example of a trivial algebra.

Last revised on April 17, 2018 at 09:53:13. See the history of this page for a list of all contributions to it.