nLab
trivial group

The trivial group is the point \bullet interpreted as a group, often denoted 11 or 00. Its underlying set is a singleton, and its unique element is the identity.

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

Given any group GG, the unique group homomorphisms from 11 to GG and from GG to 11 make 11 both a subgroup and a quotient group of GG. In such a guise, it is called the trivial subgroup or trivial quotient group of GG; the former is also called the identity subgroup.

We also denote the trivial group as {1}\{1\} or {0}\{0\}, especially when viewed as a trivial subgroup. The trivial quotient group of GG may be denoted G/GG/G or {G}\{G\}.

The trivial group is an example of a trivial algebra.

Revised on November 23, 2011 11:22:58 by Toby Bartels (71.31.209.116)