Multiplicative group

Concepts in group theory
category of groups
subgroups, normal subgroups
group homomorphisms, kernel, image, quotient
direct product, direct sum
semidirect product, wreath product
Types of groups
simple, finite, infinite
discrete, continuous
multiplicative, additive
cyclic, abelian, dihedral
nilpotent, solvable
list of group theory topics
glossary of group theory

In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context

• any group $\scriptstyle\mathfrak{{G}} \,\!$ whose binary operation is written in multiplicative notation (instead of being written in additive notation as usual for abelian groups),
• the underlying group under multiplication of the invertible elements of a field, ring, or other structure having multiplication as one of its operations. In the case of a field F, the group is {F - {0}, •}, where 0 refers to the zero element of the F and the binary operation • is the field multiplication,
• the algebraic torus $\scriptstyle\mathbf{GL}_1$.

Group scheme of roots of unity

The group scheme of $n$-th roots of unity is by definition the kernel of the $n$-power map on the multiplicative group $\scriptstyle\mathbf{GL}_1$, considered as a group scheme. That is, for any integer $n>1$ we can consider the morphism on the multiplicative group that takes $n$-th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism $e$ that serves as the identity.

The resulting group scheme is written $\mu_n$. It gives rise to a reduced scheme, when we take it over a field $\scriptstyle\mathbb{K}$, if and only if the characteristic of $\scriptstyle\mathbb{K}$ does not divide $n$. This makes it a source of some key examples of non-reduced schemes (schemes with nilpotent elements in their structure sheaves); for example $\mu_p$ over a finite field with $p$ elements for any prime number $p$.

This phenomenon is not easily expressed in the classical language of algebraic geometry. It turns out to be of major importance, for example, in expressing the duality theory of abelian varieties in characteristic $p$ (theory of Pierre Cartier). The Galois cohomology of this group scheme is a way of expressing Kummer theory.

References

• Michiel Hazewinkel, Nadiya Gubareni, Nadezhda Mikhaĭlovna Gubareni, Vladimir V. Kirichenko. Algebras, rings and modules. Volume 1. 2004. Springer, 2004. ISBN 1-4020-2690-0

See also

• multiplicative group of integers modulo n
• additive group