Introduction to Multiplicative Group

What is/are Multiplicative Group?

Multiplicative Group - We consider the problem of testing whether the points in a complex or real variety with non-zero coordinates form a multiplicative group or, more generally, a coset of a multiplicative group. ^[1] In this paper, the multiplicative group F q ∗ = F q ∖ { 0 } is decomposed into mutually disjoint union of gcd ( 2 K l m , q − 1 ) cosets of the cyclic group generated by ξ 2 K l m p n , where ξ is a primitive element of F q. ^[2] This group is isomorphic to multiplicative group formed by three cubic roots of unity. ^[3] In this paper, we make another connection by constructing and studying digraphs whose vertices are the elements of the multiplicative group of the finite fields \(\mathbb{Z}_{p}\) for certain primes \(p\). ^[4] The new approach to study this problem implies building of a multiplicative group under multiplication between no-failure (failure) probability rates as a number of units of the probability measure per unit of external (physical) time and the rate of functional (internal) time as the amount of external (physical) time per unit of the probability measure. ^[5] Let T 4 = { ± 1 , ± i } be the subgroup of fourth roots of unity inside T , the multiplicative group of complex units. ^[6] We present an upper bound on the number of solutions of an algebraic equation $$P(x,y)=0$$P(x,y)=0 where x and y belong to the union of cosets of some subgroup of the multiplicative group $$\kappa ^*$$κ∗ of some field of positive characteristic. ^[7] Our algorithm does not depend on the heuristic of smooth numbers, and it will find a primitive element of multiplicative group K. ^[8] We will show that this set forms a multiplicative group. ^[9] In the Beltrami-Klein model of the Lobachevskii space, we obtain explicit expressions for the real multiplicative group actions preserving hyperbolic pencil of straight lines, and for the real additive group actions preserving parabolic pencil of straight lines. ^[10] A system is called equivariant if it is invariant with respect to a set of coordinate transformations associated to the elements of a multiplicative group. ^[11] In this work we generalize the results of [9] to the higher level case: we define n-th root selections in fields of characteristic ≠ 2, that is subgroups of the multiplicative group of a field whose existence is equivalent to the existence of a partial inverse of the x ↦ xn function, provide necessary and sufficient conditions for such a subgroup to exist, study their existence under field extensions, and give some structural results describing the behaviour of maximal n-th root selection fields. ^[12] Let D be a division ring with center F and assume that N is a locally soluble almost subnormal subgroup of the multiplicative group D∗ of D. ^[13] The logarithmic multiplicative group is a proper group object in logarithmic schemes, which morally compactifies the usual multiplicative group. ^[14] A complex unit gain graph (or $\mathbb{T}$-gain graph) is a triple $\Phi=(G, \mathbb{T}, \varphi)$ ($(G, \varphi)$ for short) consisting of a graph $G$ as the underlying graph of $(G, \varphi)$, $\mathbb{T}= \{ z \in C:|z|=1 \} $ is a subgroup of the multiplicative group of all nonzero complex numbers $\mathbb{C}^{\times}$ and a gain function $\varphi: \overrightarrow{E} \rightarrow \mathbb{T}$ such that $\varphi(e_{ij})=\varphi(e_{ji})^{-1}=\overline{\varphi(e_{ji})}$. ^[15] It is shown that for a group $$\mathbb{U}_\mathbb{K}$$ of algebraic units of the ring of algebraic integers of $$\mathbb{Z}_\mathbb{K}$$ an algebraic field $$\mathbb{K}$$ the set of $$\mathbb{D}(M)_{\mathbb{U}_\mathbb{K}}$$ of entire Dirichlet series, $$a(1)\in\mathbb{U}_\mathbb{K}$$, is multiplicative group. ^[16] We investigate the Loewy structure of the fixpoint algebra of the group algebra of the additive group of a finite field F under the action of a subgroup of the multiplicative group of F. ^[17] In this paper, we use the theory of multiplicative group of integers modulo n to establish a dual-projection security model. ^[18] In any of the three cases, whether q is even, or $$q\equiv 3 { \text{ mod } }4$$ , or $$q\equiv 1 { \text{ mod } }4$$ , we use a decomposition of the multiplicative group of E in order to determine a (canonical) partition of the point set of $$\Gamma $$ into $$q+1$$ ovoids. ^[19] Let $mathcal{S}_q$ denote the group of all square elements in the multiplicative group $mathbb{F}_q^*$ of a finite field $mathbb{F}_q$ of odd characteristic containing $q$ elements‎. ^[20] The algebraic algorithm used to synthesize S-box basically exploits one–one correspondence between the multiplicative group of units of the local ring $${\mathbb{Z}}_{512}$$Z512 and the Galois field $$\varvec{F}_{256}$$F256. ^[21] A ring is, in particular, an additive group, and its invertible elements form a multiplicative group. ^[22] In this case, the modular homomorphism m : Γ → R+ into the multiplicative group R+ of positive real numbers is defined by the formula m(γ) = [E : E ∩ γEγ−1][γEγ−1 : E ∩ γEγ−1]−1 for γ ∈ Γ. ^[23] Using our previous results on the enumeration of Hopf-Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group $M$ and additive group $A$. ^[24] A complex unit gain graph (or ${\mathbb T}$-gain graph) is a triple $\Phi=(G, {\mathbb T}, \varphi)$ (or $(G, \varphi)$ for short) consisting of a simple graph $G$, as the underlying graph of $(G, \varphi)$, the set of unit complex numbers $\mathbb{T}= \{ z \in C:|z|=1 \}$ % is a subgroup of the multiplicative group of all nonzero complex numbers $\mathbb{C}^{\times}$ and a gain function $\varphi: \overrightarrow{E} \rightarrow \mathbb{T}$ with the property that $\varphi(e_{i,j})=\varphi(e_{j,i})^{-1}$. ^[25] Pairing is a map from two additive rational point groups $\mathbb{G}_{1}, \mathrm{G}_{2}$ to a multiplicative group $\mathbb{G}_{3}$, however, it requires complexity computation. ^[26] Let g be a generator of a multiplicative group $${\mathbb {G}}$$ G. ^[27] Notably, the approach based on decompositions of additive or multiplicative groups of finite fields turns out to be a very successful one in constructing OAI functions, where some original ideas are contributed by Tu and Deng (2012), Tang, et al. ^[28] Motivated by some algorithmic problems, we give lower bounds on the size of the multiplicative groups containing rational function images of low-dimensional affine subspaces of a finite field~$\mathbb{F}_{q^n}$ considered as a linear space over a subfield $\mathbb{F}_q$. ^[29] More importantly, discrete logarithm values are significantly different under different generators, and the multiplicative group adopted in the proposed algorithm has as many as 128 generators. ^[30] In the first three sections, we develop some basic facts about hypergeometric sheaves on the multiplicative group ${\mathbb G}_m$ in characteristic $p >0$. ^[31] We study the Galois symbol map associated to the multiplicative group and an abelian variety which has good ordinary reduction over a $p$-adic field. ^[32] Let $F$ be a local non-archimedian field of positive characteristic, $D$ be a skew-field with center $F$ and $ G=D^{\star}$ be the multiplicative group of $D$. ^[33] In terms of neural processes that might support computationally complex behavior, our hypothesis suggests that we should look for evidence of 2 operations and for symmetries corresponding to the additive and multiplicative groups. ^[34] Let $\lambda_1(n)$ denote the least invariant factor in the invariant factor decomposition of the multiplicative group $M_n = (\mathbb Z/n\mathbb Z)^\times$. ^[35]