Multiplicative Group

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 g·h or gh. In a multiplicative group, the identity element is denoted 1, and the inverse of the element g is written as g^(-1), voiced "g 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 {1} and C_2={-1,1}, the latter of which is isomorphic to the cyclic additive group C_2. The elements of C_2 are the square roots of unity, and in general, the set C_n of all complex nth roots of unity is a cyclic multiplicative group of order n,


where the generator omega is any primitive nth root of unity. These groups are all subgroups of the multiplicative group C^*=C\{0}, formed by all nonzero complex numbers. In general, if K is a division algebra, then the set K^*=K\{0} is always a multiplicative group, which is commutative iff K is a field. If K is the Galois field GF(p^n), its multiplicative group is always cyclic. More generally, the invertible elements of a unit ring A form a multiplicative group, which is usually denoted U(A) or A^*. The invertible elements of the ring Z_n are the residue classes of all elements a in {1,...,n-1} which are coprime with respect to n. The group U(Z_n) obtained in this way has phi(n) elements, where phi(n) denotes the totient function.

The set GL(n,K) of all nonsingular n×n matrices with entries in the field K is a multiplicative group with respect to matrix multiplication, called the general linear group of order n over K. It has the special linear group SL(n,K) as a subgroup. If K=R (or K=C) we can also consider the orthogonal group O(n,R), the special orthogonal group SO(n,R), (the unitary group U(n,C) and the special unitary group SU(n,C)) as multiplicative subgroups of GL(n,R) (or GL(n,C)). Other subgroups are formed by the following sets:

 {scalar matrices}^* subset= {diagonal matrices}^* subset= {upper triangular matrices}^*,

where the symbol ^* means that we consider only the matrices in which all diagonal elements are nonzero.

The quotient of a multiplicative group G with respect to a normal subgroup H is a multiplicative group with respect to the coset product defined by


An example is the projective general linear group

 PGL(n,K)=GL(n,K)/{scalar matrices}^*.

The name multiplicative group is also applied to groups of maps, where the operation is the map composition  degrees. 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 n, the nth power of a map f is defined as

 f^n=f degreesf degrees... degreesf_()_(n times).

Negative powers are also defined as usual, so


if n<0.

See also

Additive Group, Multiplication, Multiplication Table, Multiplicative Inverse, Quaternion Group

This entry contributed by Margherita Barile

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Barile, Margherita. "Multiplicative Group." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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