Trivial Group
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 
.
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trivial group