Trivial Group

The trivial group, denoted E or <e>, sometimes also called the identity group, is the unique (up to isomorphism) group containing exactly one element e, the identity element. Examples include the zero group (which is the singleton set {0} with respect to the trivial group structure defined by the addition 0+0=0), the multiplicative group {1} (where 1·1=1), the point group C_1, and the integers modulo 1 under addition. When viewed as a permutation group on p letters, the trivial group E_p consists of the single element which fixes each letter.

The trivial group is (trivially) Abelian and cyclic.

The multiplication table for <e> is given below.


The trivial group has the single conjugacy class {1} and the single subgroup {1}.

See also

Cyclic Group, Finite Group, Group, Identity Element, Zero Group

Portions of this entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd and Weisstein, Eric W. "Trivial Group." From MathWorld--A Wolfram Web Resource.

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