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...

{\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...

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)...

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

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

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...

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...

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

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...

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

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...

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

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...

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

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...

matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible...

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...

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...

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

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

+ (−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...

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

denotes the direct product and U(1), known as the circle group, is the multiplicative group of all complex numbers with absolute value 1. For completeness...

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...

of two group members. Thus, the multiplicative group of rational functions can be factored into the multiplicative group of complex numbers (the associated...