ABSTRACT In this paper we give a polynomial-time quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomial-time quantum algorithms as well. Our algorithm works in the setting of black-box groups, wherein none of these problems have polynomial-time classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.

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This paper introduces new techniques for the efficient computation of a Fourier transform on a finite group. We present a divide and conquer approach to the computation. The divide aspect uses factorizations of group elements to reduce the matrix sum of products for the Fourier transform to simpler sums of products. This is the separation of variables algorithm. The conquer aspect is the final computation of matrix products which we perform efficiently using a special form of the matrices. This form arises from the use of subgroup-adapted representations and their structure when evaluated at elements which lie in the centralizers of subgroups in a subgroup chain. We present a detailed analysis of the matrix multiplications arising in the calculation and obtain easy-to-use upper bounds for the complexity of our algorithm in terms of representation theoretic data for the group of interest. Our algorithm encompasses many of the known examples of fast Fourier transforms. We recover the b...

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This note describes recent research using structural properties of finite groups to devise efficient algorithms for group computation.

l over the generators grows as c l for some constant c>1 depending on G. For groups with abelian subgroups of finite index, we obtain a Las Vegas algorithm for several basic computational tasks, including membership testing and computing a presentation. This generalizes recent work of Beals and Babai, who give a Las Vegas algorithm for the case of finite groups, as well as recent work of Babai, Beals, Cai, Ivanyos, and Luks, who give a deterministic algorithm for the case of abelian groups. # 1999 Academic Press Article ID jcss.1998.1614, available online at http:##www.idealibrary.com on 260 0022-0000#99 #30.00 Copyright # 1999 by Academic Press All rights of reproduction in any form reserved. * Research conducted while visiting IAS and DIMACS and supported in part by an NSF Mathematical Sciences

Let B = fBmgm>0 be a countable family of nite groups whose elements are uniquely encoded as strings of uniform length, and group operations are computable in time bounded by a polynomial in m. A black-box group over a group family B is a subgroup of some member of B and is presented by a generator set. In this paper we study the complexity of several algorithmic problems for abelian black-box groups and solvable black-box groups. We design a suitable oracle algorithm that computes an independent set of generators for a given abelian black-box group. Using this we show that the problems of Membership Testing, Group Intersection, Order Verication, and Group Isomorphism over abelian blackbox groups are in SPP. We also show that Group Factorization, Coset Intersection, and Double Coset Membership problems over abelian black-box groups are in LWPP. As a consequence, all these problems are low for PP and C=P. We dene the notion of canonical generator sets for classes of groups ...

) IGOR PAK , SERGEY BRATUS y 1 Introduction Let G be a finite group. A sequence of k group elements (g1 ; : : : ; gk ) is called a generating k-tuple of G if the elements generate G (we write hg1 ; : : : ; gk i = G). Let Nk (G) be the set of all generating k-tuples of G, and let Nk (G) = jNk (G)j. We consider two related problems on generating k-tuples. Given G and k ? 0, 1) Determine Nk (G) 2) Generate random element of Nk (G), each with probability 1=Nk (G) The problem of determining the structure of Nk (G) is of interest in several contexts. The counting problem goes back to Philip Hall, who expressed Nk (G) as a Mobius type summation of Nk (H) over all maximal subgroups H ae G (see [23]). Recently the counting problem has been studied for large simple groups where remarkable progress has been made (see [25, 27]). In this paper we analyze Nk for solvable groups and products of simple groups. The sampling problem, while often used in theory as a tool for approximate counting...

Let F be a field and t an indeterminate. In this paper we consider aspects of the problem of deciding if a finitely generated subgroup of GL(n, F(t)) is finite. When F is a number field, the analysis may be easily reduced to deciding finiteness for subgroups of GL(n, F), for which the results of [1] can be applied. When F is a finite field, the situation is more subtle. In this case our main results are a structure theorem generalizing a theorem of Weil and upper bounds on the size of a finite subgroup generated by a fixed number of generators with examples of constructions almost achieving the bounds. We use these results to then give exponential deterministic algorithms for deciding finiteness as well as some preliminary results towards more e#cient randomized algorithms. 1 Introduction Recently, computational group theory has directed increased attention to the development of algorithms for studying matrix groups. In particular, various recognition algorithms are of importance. The...