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Outside of number theory the simpler notation is often used, though it can be confused with the p-adic integers when n is a prime number. The multiplicative group of integers modulo n, which is the group of units in this ring, may be written as (depending on the author) (/), (/), (/), (/) (for German Einheit, which translates as unit), , or ...
The circle group is more than just an abstract algebraic object. It has a natural topology when regarded as a subspace of the complex plane. Since multiplication and inversion are continuous functions on , the circle group has the structure of a topological group.
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 n th roots of unity form under multiplication a cyclic group of order n, and in fact these groups comprise all of the finite subgroups of the multiplicative group of the complex number field. A generator for this cyclic group is a primitive n th root of unity. The n th roots of unity form an irreducible representation of any cyclic group of ...
A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication.
n, and is called the group of units modulo n, or the group of primitive classes modulo n. As explained in the article multiplicative group of integers modulo n, this multiplicative group (× n) is cyclic if and only if n is equal to 2, 4, p k, or 2 p k where p k is a power of an odd prime number.
If p is a prime number, then any group with p elements is isomorphic to the simple group Z/pZ. A number n is called a cyclic number if Z/nZ is the only group of order n, which is true exactly when gcd(n, φ(n)) = 1. [13] The sequence of cyclic numbers include all primes, but some are composite such as 15. However, all cyclic numbers are odd ...
The quotient group / is isomorphic to the circle group, the group of complex numbers of absolute value 1 under multiplication, or correspondingly, the group of rotations in 2D about the origin, that is, the special orthogonal group .
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