We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Zp , whenever p is a fixed prime. For the induction step, we introduce the ...

Let S be a 2-group. The rank (normal rank) of S is the maximal dimension of an elementary abelian subgroup (a normal elementary abelian subgroup) of S over Z2. The purpose of this article is to determine the rank and normal rank of S, where S is a Sylow 2-subgroup of the classical groups of odd characteristic.

For each locally compact non-compact c-compact Abelian group G whose dual is an I-group and each compact Abelian group ,?J such that there is a one-to-one continous homomorphism from G onto a dense subgroup of Z, we show the existence of cocycles on Z whose associated unitary representations of G have purely singular continuous spectrum, as well as the existence of cocycles on Z whose associate...

We study the base sizes of finite quasiprimitive permutation groups twisted wreath type, which are precisely with a unique minimal normal subgroup that is also non-abelian, non-simple and regular. Every group type isomorphic to product G=Tk:P acting on its Ω=Tk, where T some non-abelian simple P transitively k={1,…,k} k⩾2. prove if G primitive Ω k, then has size 2. proportion pairs points bases...

We generalize the classical Post correspondence problem (PCPn) and its non-homogeneous variation (GPCPn) to non-commutative groups and study the computational complexity of these new problems. We observe that PCPn is closely related to the equalizer problem in groups, while GPCPn is connected to the double twisted conjugacy problem for endomorphisms. Furthermore, it is shown that one of the str...

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Zp , whenever p is a fixed prime. For the induction step, we introduce the ...

let $g$ be a finite group‎. ‎a subset $x$ of $g$ is a set of pairwise non-commuting elements‎ ‎if any two distinct elements of $x$ do not commute‎. ‎in this paper‎ ‎we determine the maximum size of these subsets in any finite‎ ‎non-abelian metacyclic $2$-group and in any finite non-abelian $p$-group with an abelian maximal subgroup‎.

Motivated by quantum simulation, we consider lattice Hamiltonians for Yang-Mills gauge theories with finite group, example a subgroup of compact Lie group. We show that the electric Hamiltonian admits an interpretation as certain natural, nonunique Laplacian operator on Abelian or non-Abelian group and derive some consequences from this fact. Independent chosen Hamiltonian, provide full explici...

A group G is said to have restricted centralizers if for each g in the centralizer $$C_G(g)$$ either finite or has index G. Shalev showed that a profinite with virtually abelian. Given set of primes $$\pi $$ , we take interest groups -elements. It shown such an open subgroup form $$P\times Q$$ where P abelian pro- and Q '$$ subgroup. This significantly strengthens result from our earlier paper.

We show that for every integer d ∈ N, there is N(d) ∈ N such that if K is any field and F is a finite subset of GLd(K), which generates a non amenable subgroup, then F contains two elements, which freely generate a non abelian free subgroup. This improves the original statement of the Tits alternative. It also implies a growth gap and a co-growth gap for non-amenable linear groups, and has cons...