Given any group , the unique group homomorphisms from to and from to make both a subgroup and a quotient group of . In such a guise, it is called the trivial subgroup or trivial quotient group of ; the former is also called the identity subgroup.
We also denote the trivial group as or , especially when viewed as a trivial subgroup. The trivial quotient group of may be denoted or .
The trivial group is an example of a trivial algebra.