Polynomial-time theory of matrix groups
Proceedings of the forty-first annual ACM symposium on Theory of computing
An efficient quantum algorithm for the hidden subgroup problem in nil-2 groups
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Quantum algorithms for a set of group theoretic problems
ICTCS'05 Proceedings of the 9th Italian conference on Theoretical Computer Science
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We consider algorithms in finite groups, given by a list of generators. We give polynomial time Las Vegas algorithms (randomized, with guaranteed correct output) for basic problems for finite matrix groups over the rationals (and over algebraic number fields): testing membership, determining the order, finding a presentation (generators and relations), and finding basic building blocks: center, composition factors, and Sylow subgroups. These results extend previous work on permutation groups into the potentially more significant domain of matrix groups. Such an extension has until recently been considered intractable. In case of matrix groups G of characteristic p, there are two basic types of obstacles to polynomial-time computation: number theoretic (factoring, discrete log) and large Lie-type simple groups of the same characteristic p involved in the group. The number theoretic obstacles are inherent and appear already in handling abelian groups. They can be handled by moderately efficient (subexponential) algorithms. We are able to locate all the nonabelian obstacles in a normal subgroup N and solve all problems listed above for G/N.