 TOPICS # 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 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 .

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. https://mathworld.wolfram.com/MultiplicativeGroup.html