# Multiplicative group

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{{Short description|Mathematical structure with multiplication as its operation}}
{{Group theory sidebar |Basics}}

In [mathematics](/source/mathematics) and [group theory](/source/group_theory), the term '''multiplicative group''' refers to one of the following concepts:
* the '''[group](/source/group_(mathematics)) under multiplication''' of the [invertible](/source/invertible) elements of a [field](/source/field_(mathematics)),{{sfn|ps=|Hazewinkel|Gubareni|Gubareni|Kirichenko|2004|p=2}} [ring](/source/ring_(mathematics)), or other structure for which one of its operations is referred to as multiplication. In the case of a field ''F'', the group is {{nowrap|(''F'' ∖ {{mset|0}}, •)}}, where 0 refers to the [zero element](/source/zero_element) of ''F'' and the [binary operation](/source/binary_operation) • is the field [multiplication](/source/multiplication),
* the [algebraic torus](/source/algebraic_torus) GL(1).

== Examples ==
* The [multiplicative group of integers modulo ''n''](/source/multiplicative_group_of_integers_modulo_n) is the group under multiplication of the invertible elements of <math>\mathbf{Z}/n\mathbf{Z}</math>.  When ''n'' is not prime, there are elements other than zero that are not invertible.
* The multiplicative group of [positive real numbers](/source/positive_real_numbers) <math>\mathbf{R}^+</math> is an [abelian group](/source/abelian_group) with 1 its [identity element](/source/identity_element). The [logarithm](/source/logarithm) is a [group isomorphism](/source/group_isomorphism) of this group to the [additive group](/source/additive_group) of real numbers, <math>\mathbf{R}</math>.
* The multiplicative group of a field <math>F</math> is the set of all nonzero elements: <math>F^\times = F \smallsetminus \{0\}</math>, under the multiplication operation. If <math>F</math> is [finite](/source/finite_field) of order ''q'' (for example {{nowrap|1=''q'' = ''p''}} a prime, and <math>F = \mathbb F_p=\mathbf{Z}/p\mathbf{Z}</math>), then the [multiplicative group](/source/finite_field) is cyclic: <math>F^\times \cong \mathrm{C}_{q-1}</math>.

== Group scheme of roots of unity ==
The '''group scheme of ''n''th [roots of unity](/source/roots_of_unity)''' is by definition the kernel of the ''n''-power map on the multiplicative group GL(1), considered as a [group scheme](/source/group_scheme). That is, for any integer {{nowrap|''n'' > 1}} we can consider the morphism on the multiplicative group that takes ''n''th powers, and take an appropriate [fiber product of schemes](/source/fiber_product_of_schemes), with the morphism ''e'' that serves as the identity.

The resulting group scheme is written ''μ''<sub>''n''</sub> (or <math>\mu\!\!\mu_n</math>{{sfn|ps=|Milne|1980|pp=xiii,66}}). It gives rise to a [reduced scheme](/source/reduced_scheme), when we take it over a field ''K'', [if and only if](/source/if_and_only_if) the [characteristic](/source/characteristic_(field)) of ''K'' does not divide ''n''. This makes it a source of some key examples of non-reduced schemes (schemes with [nilpotent element](/source/nilpotent_element)s in their [structure sheaves](/source/structure_sheaf)); for example ''μ''<sub>''p''</sub> over a [finite field](/source/finite_field) with ''p'' elements for any [prime number](/source/prime_number) ''p''. 

This phenomenon is not easily expressed in the classical language of algebraic geometry. For example, it turns out to be of major importance in expressing the [duality theory of abelian varieties](/source/duality_theory_of_abelian_varieties) in characteristic ''p'' (theory of [Pierre Cartier](/source/Pierre_Cartier_(mathematician))). The [Galois cohomology](/source/Galois_cohomology) of this group scheme is a way of expressing [Kummer theory](/source/Kummer_theory).

== See also ==
* [Multiplicative group of integers modulo n](/source/Multiplicative_group_of_integers_modulo_n)
* [Additive group](/source/Additive_group)

== Notes ==
{{reflist}}

== References ==
* {{citation |first1=Michiel |last1=Hazewinkel |author-link1=Michiel Hazewinkel |first2=Nadiya |last2=Gubareni |first3=Nadezhda Mikhaĭlovna |last3=Gubareni |first4=Vladimir V. |last4=Kirichenko |title=Algebras, rings and modules |volume=1 |year=2004 |publisher=Springer |isbn=1-4020-2690-0 }}
* {{cite book |last1=Milne |first1=James S. |author-link1=James Milne (mathematician) |title=Étale cohomology |publisher=Princeton University Press |year=1980 }}

{{DEFAULTSORT:Multiplicative Group}}
Category:Algebraic structures
Category:Group theory
Category:Field theory

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Adapted from the Wikipedia article [Multiplicative group](https://en.wikipedia.org/wiki/Multiplicative_group) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Multiplicative_group?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.