Multiplicative group of integers modulo n information
Group of units of the ring of integers modulo n
In modular arithmetic, the integers coprime (relatively prime) to n from the set of n non-negative integers form a group under multiplication modulo n, called the multiplicative group of integers modulo n. Equivalently, the elements of this group can be thought of as the congruence classes, also known as residues modulo n, that are coprime to n.
Hence another name is the group of primitive residue classes modulo n.
In the theory of rings, a branch of abstract algebra, it is described as the group of units of the ring of integers modulo n. Here units refers to elements with a multiplicative inverse, which, in this ring, are exactly those coprime to n.
Algebraic structure → Group theory Group theory
Basic notions
Subgroup
Normal subgroup
Quotient group
(Semi-)direct product
Group homomorphisms
kernel
image
direct sum
wreath product
simple
finite
infinite
continuous
multiplicative
additive
cyclic
abelian
dihedral
nilpotent
solvable
action
Glossary of group theory
List of group theory topics
Finite groups
Cyclic group Zn
Symmetric group Sn
Alternating group An
Dihedral group Dn
Quaternion group Q
Cauchy's theorem
Lagrange's theorem
Sylow theorems
Hall's theorem
p-group
Elementary abelian group
Frobenius group
Schur multiplier
Classification of finite simple groups
cyclic
alternating
Lie type
sporadic
Discrete groups
Lattices
Integers ()
Free group
Modular groups
PSL(2, )
SL(2, )
Arithmetic group
Lattice
Hyperbolic group
Topological and Lie groups
Solenoid
Circle
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Euclidean E(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n)
G2
F4
E6
E7
E8
Lorentz
Poincaré
Conformal
Diffeomorphism
Loop
Infinite dimensional Lie group
O(∞)
SU(∞)
Sp(∞)
Algebraic groups
Linear algebraic group
Reductive group
Abelian variety
Elliptic curve
v
t
e
This quotient group, usually denoted , is fundamental in number theory. It is used in cryptography, integer factorization, and primality testing. It is an abelian, finite group whose order is given by Euler's totient function: For prime n the group is cyclic, and in general the structure is easy to describe, but no simple general formula for finding generators is known.
and 24 Related for: Multiplicative group of integers modulo n information
{n}}}. In other words, the multiplicative order of a modulon is the order of a in the multiplicativegroupof the units in the ring of the integers modulo...
Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplicationof complex...
{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{\times }} , the multiplicativegroupofintegersmodulon, with the isomorphism given by a ¯ ↦ σ a ∈ G , σ a ( x...
to the additive group of Z/nZ, the integersmodulon. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every...
temperament Circle of fifths text table Pitch constellation MultiplicativegroupofintegersmodulonMultiplication (music) Circle of thirds Michael Pilhofer...
the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the number...
N ) = ( nN ) ( g N ) = ( n g ) N {\displaystyle gN=(eg)N=(eN)(gN)=(nN)(gN)=(ng)N} . Now, g N = ( n g ) N ⇔ N = ( g − 1 n g ) N ⇔ g − 1 n g ∈ N , ∀ n...
totient function, the multiplicativegroupofintegersmodulon for n = 9 and 10, respectively. This triples and doubles the number of automorphisms compared...
is the multiplicative order of 2 modulo 5k, which is φ(5k) = 4 × 5k−1 (see Multiplicativegroupofintegersmodulon).[citation needed] (sequence A140300...
examples of finite fields are the fields of prime order: for each prime number p, the prime field of order p may be constructed as the integersmodulo p, Z...
except zero modulo p has a multiplicative inverse. This is not true for composite moduli.) Following this convention, the multiplicative inverse of a residue...
number theory, a kth root of unity modulon for positive integers k, n ≥ 2, is a root of unity in the ring ofintegersmodulon; that is, a solution x to...