Combinatorica
Polynomial time solutions of some problems of computational algebra
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
Trading group theory for randomness
STOC '85 Proceedings of the seventeenth annual ACM symposium on Theory of computing
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Fast Monte Carlo algorithms for permutation groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Local expansion of vertex-transitive graphs and random generation in finite groups
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Nearly linear time algorithms for permutation groups with a small base
ISSAC '91 Proceedings of the 1991 international symposium on Symbolic and algebraic computation
Bounded round interactive proofs in finite groups
SIAM Journal on Discrete Mathematics
Computations for algebras and group representations
Computations for algebras and group representations
Deciding finiteness of matrix groups in deterministic polynomial time
ISSAC '93 Proceedings of the 1993 international symposium on Symbolic and algebraic computation
On deciding finiteness of matrix groups
Journal of Symbolic Computation
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Let G be a group of matriceswith integer entries, given by a list of generators. It is known thatmembership in such a group is undecidable, even for 4 x 4 integralmatrices [Mi].In this paper we show that one can decide whether or notG isfinite, in Las Vegas polynomialtime.The key estimate derived makes the entire “black boxgroup” theory ([BSz], [BCFLS], [Ba2], [BKL]) applicable to thefinite groups of integral matrices. In particular it follows that inthis case, structural properties such assolvability andnilpotence are decidable in MonteCarlo polynomial time; and membership, order,isomorphism, and a host of other problems are in therelatively low complexity class AM∩coAM [Ba1].We give two algorithms. The simpler one (Monte Carlo but not LasVegas) employs a refinement of the randomwalk technique over groups, developed in [Ba3](applied here to infinite groups). The termination rule rests on a newestimate on the bit-size of the elements of finite groupsG, obtained via polynomial timesymbolic manipulation of representations over algebraicnumber fields using results of [BR]. This symbolicmanipulation technique is the basis of the Las Vegas algorithm. (A LasVegas algorithm is a rnadomized algorithm which never errs; but withsmall probability, it may r eport failure.)