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trivial group
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Contents
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 $1$ , while in the context of abelian groups it is usually denoted $0$ .

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 $1 \hookrightarrow G$ is the unique such group homomorpism.

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