Multiplicative group

Mathematical structure with multiplication as its operation In mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referred to as multiplication. In the case of a field F, the group is (F ∖ {0}, •), where 0 refers to the zero element of F and the binary operation • is the field multiplication, the algebraic torus GL(1). Examples The multiplicative group of integers modulo n is the group under multiplication of the invertible elements of Z / n Z {\displaystyle \mathbf {Z} /n\mathbf {Z} } . When n is not prime, there are elements other than zero that are not invertible. The multiplicative group of positive real numbers R + {\displaystyle \mathbf {R} ^{+}} is an abelian group with 1 as its identity element. The logarithm is a group isomorphism of this group to the additive group of real numbers, R {\displaystyle \mathbf {R} } . The multiplicative group of a field F {\displaystyle F} is the set of all nonzero elements: F × = F ∖ { 0 } {\displaystyle F^{\times }=F\smallsetminus \{0\}} , under the multi