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Concept
Multiplicative group
Summary
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)..
The multiplicative group of integers modulo n is the group under multiplication of the invertible elements of . When n is not prime, there are elements other than zero that are not invertible.
The multiplicative group of positive real numbers is an abelian group with 1 its identity element. The logarithm is a group isomorphism of this group to the additive group of real numbers, .
The multiplicative group of a field is the set of all nonzero elements: , under the multiplication operation. If is finite of order q (for example q = p a prime, and ), then the multiplicative group is cyclic: .
The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group 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 of schemes, with the morphism e that serves as the identity.
The resulting group scheme is written μn (or ). It gives rise to a reduced scheme, when we take it over a field K, if and only if the characteristic of 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 μp 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. For example, it turns out to be of major importance in expressing the duality theory of abelian varieties in characteristic p (theory of Pierre Cartier).
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Related concepts (16)
Field with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F1 refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects.
Additive group
An additive group is a group of which the group operation is to be thought of as addition in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structures equipped with several operations for specifying the structure obtained by forgetting the other operations. Examples include the additive group of the integers, of a vector space and of a ring. This is particularly useful with rings and fields to distinguish the additive underlying group from the multiplicative group of the invertible elements.
Finite ring
In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements. Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history. Although rings have more structure than groups, the theory of finite rings is simpler than that of finite groups.
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