The trivial group, denoted 
or 
,
sometimes also called the identity group, is the unique (up to isomorphism) group
containing exactly one element 
, the identity element.
Examples include the zero group (which is the singleton
set 
with respect to the trivial group structure defined by the addition 
), the multiplicative
group 
(where 
),
the point group 
, and the integers modulo 1 under addition. When viewed as
a permutation group on 
letters, the trivial group 
consists of the single element which fixes each letter.
The trivial group is (trivially) Abelian and cyclic.
The multiplication table for 
is given below.
 | 1 |
| 1 | 1 |
The trivial group has the single conjugacy class 
and the single subgroup 
.
