mathematics and group theory, the term multiplicative group refers to one of the following concepts: the group under multiplication of the invertible...

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...

Multiplicative may refer to: Multiplication Multiplicative function Multiplicative group Multiplicative identity Multiplicative inverse Multiplicative...

spectrum of a group ring. More generally, one can form groups of multiplicative type by letting A be a non-constant sheaf of abelian groups on S. For a...

Z {\displaystyle \mathbb {Z} } has a (multiplicative) inverse. The desire for the existence of multiplicative inverses suggests considering fractions...

from an orthogonal group over a field F to the quotient group F× / (F×)2 (the multiplicative group of the field F up to multiplication by square elements)...

{G} _{m}} is called the multiplicative group, because its k {\displaystyle k} -points are isomorphic to the multiplicative group of the field k {\displaystyle...

conventions for abelian groups – additive and multiplicative. Generally, the multiplicative notation is the usual notation for groups, while the additive...

element can be written as an integer power of g {\displaystyle g} in multiplicative notation, or as an integer multiple of g {\displaystyle g} in additive...

have the Klein four-group as their additive substructure. If R × {\displaystyle \mathbb {R} ^{\times }} denotes the multiplicative group of non-zero reals...

multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group...

{\begin{pmatrix}a&b\\0&1\end{pmatrix}}\mapsto a^{u}} is a group homomorphism. Consider a multiplicative group of positive real numbers (R+, ⋅) for any complex...

solution, i.e., when it exists, a modular multiplicative inverse is unique: If b and b' are both modular multiplicative inverses of a respect to the modulus...

immediately to basic properties of p-adic numbers: Addition, multiplication and multiplicative inverse of p-adic numbers are defined as for formal power...

follows, in part, since the multiplicative group of complex numbers C × {\displaystyle \mathbb {C} ^{\times }} is connected. The group manifold GL ⁡ ( n , C...

primitive element of a finite field GF(q) is a generator of the multiplicative group of the field. In other words, α ∈ GF(q) is called a primitive element...

mathematics, 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...

logarithmic derivative lie two basic facts about GL1, that is, the multiplicative group of real numbers or other field. The differential operator X d d X...

mathematics, the circle group, denoted by T {\displaystyle \mathbb {T} } or ⁠ S 1 {\displaystyle \mathbb {S} ^{1}} ⁠, is the multiplicative group of all complex...

algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups). Every...

operation instead of multiplicative notation). For a general subgroup ⁠ H {\displaystyle H} ⁠, it is desirable to define a compatible group operation on the...

generalizations See Multiplication in group theory, above, and multiplicative group, which for example includes matrix multiplication. A very general, and...

1{\pmod {n}}} . In other words, the multiplicative order of a modulo n is the order of a in the multiplicative group of the units in the ring of the integers...

arithmetic), meaning that the order of 2 {\displaystyle 2} in the multiplicative group ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} is...

n-1\right\rbrace } . The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers. Using the class...

primes between B1 and a larger limit B2. If the algebraic group is the multiplicative group mod N, the one-sided identities are recognised by computing...

R)\to R^{\times }.} where R × {\displaystyle R^{\times }} is the multiplicative group of R {\displaystyle R} (that is, R {\displaystyle R} excluding 0...

a primitive root modulo n if and only if g is a generator of the multiplicative group of integers modulo n. Gauss defined primitive roots in Article 57...

+ (−a) = 0. Multiplicative inverses: for every a ≠ 0 in F, there exists an element in F, denoted by a−1 or 1/a, called the multiplicative inverse of a...

ElGamal encryption can be defined over any cyclic group G {\displaystyle G} , like multiplicative group of integers modulo n if and only if n is 1, 2, 4...