A group whose group operation is identified with multiplication. As with normal multiplication, the multiplication operation on group elements is
either denoted by a raised dot 
or omitted entirely, giving the notation 
or 
. In a multiplicative group, the identity
element is denoted 1, and the inverse of the element 
is written as 
, voiced "
inverse." This notation and terminology is borrowed from
the multiplicative groups formed by numbers, where the operation is the usual arithmetical
product, the identity element is the number 1, and the inverse coincides with the
multiplicative reciprocal.
The simplest examples are the trivial group 
and 
, the latter of which is isomorphic to the cyclic additive group 
. The elements of 
are the square roots of unity, and in general, the set 
of all complex 
th roots of unity is a cyclic multiplicative group of order
,
 |
(1)
|
where the generator 
is any primitive 
th root of unity. These groups are all subgroups of the multiplicative
group 
,
formed by all nonzero complex numbers. In general, if 
is a division algebra,
then the set 
is always a multiplicative group, which is commutative iff 
is a field. If 
is the Galois field 
, its multiplicative group is always cyclic. More generally,
the invertible elements of a unit ring 
form a multiplicative group, which is usually denoted 
or 
. The invertible elements of the ring 
are the residue classes of all elements 
which are coprime with respect to 
. The group 
obtained in this way has 
elements, where 
denotes the totient function.
The set 
of all nonsingular 
matrices with entries in the field 
is a multiplicative group with respect to matrix multiplication,
called the general linear group of order 
over 
. It has the special linear
group 
as a subgroup. If 
(or 
)
we can also consider the orthogonal group 
, the special orthogonal group 
, (the unitary group 
and the special
unitary group 
)
as multiplicative subgroups of 
(or 
). Other subgroups are formed by the following sets:
 |
(2)
|
where the symbol 
means that we consider only the matrices in which all diagonal elements are nonzero.
The quotient of a multiplicative group 
with respect to a normal subgroup 
is a multiplicative group with respect
to the coset product defined by
 |
(3)
|
An example is the projective general linear group
 |
(4)
|
The name multiplicative group is also applied to groups of maps, where the operation is the map composition 
. This is the cases of transformation
groups (such as the rotation group) and the
symmetric groups and their subgroups (such as
the alternating groups). For all positive integers
,
the 
th
power of a map 
is defined as
 |
(5)
|
Negative powers are also defined as usual, so
 |
(6)
|
if 
.
