We give efficient quantum algorithms for HIDDEN TRANSLATION and HIDDEN SUBGROUP in a large class of non-abelian groups including solvable groups of bounded exponent and of bounded derived series. Our algorithms are recursive. For the base case, we solve efficiently HIDDEN TRANSLATION in Z p , whenever p is a fixed prime. For the induction step, we introduce the problem ORBIT COSET generalizing ...

The commutator of Lie algebra is extended to a product of fuzzy Lie subspaces over fuzzy field. An action of fuzzy field is introduced as an extension of scalar multiplication in Lie algebra. The notion of derived series and descending central series of fuzzy Lie ∼ideal over fuzzy field and the concepts of ∼nilpotent and ∼solvable fuzzy Lie ∼ideals over fuzzy field are studied. Mathematics Subj...

we describe under various conditions abelian subgroups of the automorphism‎ ‎group $mathrm{aut}(t_{n})$ of the regular $n$-ary tree $t_{n}$‎, ‎which are‎ ‎normalized by the $n$-ary adding machine $tau =(e‎, ‎dots‎, ‎e,tau )sigma _{tau‎ ‎}$ where $sigma _{tau }$ is the $n$-cycle $left( 0,1‎, ‎dots‎, ‎n-1right) $‎. ‎as‎ ‎an application‎, ‎for $n=p$ a prime number‎, ‎and for $n=4$‎, ‎we prove that...

In 1963 Gasch utz proved the existence in nite solvable groups of certain characteristic conjugacy classes of subgroups associated with formations. His initial work in 6] has developed into an extensive theory. In this paper we introduce a new approach to this theory and theoretical results that yield practical methods to compute formation theoretic subgroups such as F-normalizers and F-coverin...

We introduce the notion of a powerfully solvable group. These are powerful groups possessing an abelian series special kind. include in particular class nilpotent groups. will also see that for certain rich we can naturally term simple group and prove Jordan-Hölder type theorem justifies term.

The first and second Born approximation are studied with the path integral representation for T matrix. The T matrix is calculated for Woods-Saxon potential scattering. To make corresponding integrals solvable analytically, an approximate function for the Woods-Saxon potential is used. Finally it shown that the Born series is converge at high energies and orders higher than two in Born approxim...

Every finite group G $G$ has a normal series each of whose factors is either solvable or direct product non-abelian simple groups. The minimum number nonsolvable factors, attained on all possible such in , called the length ? ( ) $\lambda (G)$ . In present paper, we prove theorem about permutation representations groups fixed length. As consequence, show that at least n $n$ no nontrivial word m...

the concept of the bipartite divisor graph for integer subsets has been considered in [‎graph combinator., ‎ 26 (2010) 95--105‎.]. ‎in this paper‎, ‎we will consider this graph for the set of character degrees of a finite group $g$ and obtain some properties of this graph‎. ‎we show that if $g$ is a solvable group‎, ‎then the number of connected components of this graph is at most $2$ and if $g...

We use the "tridiagonal representation approach" to solve time-independent Schr\"odinger equation for bound states of generalized versions trigonometric and hyperbolic P\"oschl-Teller potentials. These new solvable potentials do not belong conventional class exactly problems. The solutions are finite series square integrable functions written in terms Jacobi polynomial.