## Constructive recognition of a black box group isomorphic to GL(n, 2) (1997)

Venue: | IN GROUPS AND COMPUTATION II, DIMACS SERIES IN DISCRETE MATHEMATICS AND COMPUTER SCIENCE |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Cooperman97constructiverecognition,

author = {Gene Cooperman and Larry Finkelstein and Steve Linton},

title = {Constructive recognition of a black box group isomorphic to GL(n, 2)},

booktitle = {IN GROUPS AND COMPUTATION II, DIMACS SERIES IN DISCRETE MATHEMATICS AND COMPUTER SCIENCE},

year = {1997},

pages = {85--100},

publisher = {American Mathematical Society}

}

### OpenURL

### Abstract

A Monte Carlo algorithm is presented for constructing the natural representation of a group G that is known to be isomorphic to GL(n,2). The complexity parameters are the natural dimension n and the storage space required to represent an element of G. What is surprising about this result is that both the data structure used to compute the isomorphism and each invocation of the isomorphism require polynomial time complexity. The ultimate goal is to eventually extend this result to the larger question of constructing the natural representation of classical groups. Extensions of the methods developed in this paper are discussed as well as open questions.

### Citations

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(Show Context)
Citation Context ...domly chosen element of G. Ideally, one would like to have an efficient implementation of Rand() which produces elements which are uniformly distributed. In the case where G is black box group, Babai =-=[1]-=- achieves this goal. Simply put, he shows how to construct from G, a set of O(log |G|) elements, at a cost of O(log 5 |G|) group multiplications, from which nearly uniform distributed random elements ... |

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Citation Context ...s volume). It is also required that we know the prime factorization for each divisor of |G| of the form 2i −1, 1 ≤ i ≤ n. This allows us to use the Bounded Order Algorithm of Celler and Leedham-Green =-=[5]-=- for computing the order of an element x of G when it is known in advance that |x| divides 2i − 1, 1 ≤ i ≤ n. Our approach is to determine elements of G which can easily be found in a black box settin... |

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(Show Context)
Citation Context ...le to construct a set S of transvections in G satisfying P. In order to answer (2), it suffices to show that each generator g ∈ G of G is an element of G0. A recent result of Celler and Leedham-Green =-=[6]-=- seems to apply directly to this problem. We first compute Θ(g) and then apply [6] to express Θ(g) as word in Θ(S). By shadowing the computation in G, we can then produce an element g0 ∈ G0 such that ... |

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(Show Context)
Citation Context ...ples of black box groups are matrix groups and permutation groups. This effort was initially motivated by recent efforts to identify the structure of matrix groups defined over finite fields (see [3] =-=[7]-=- and [9] in this volume). However, an important advantage of working in a black box setting is that the techniques rely only on the structure of the group and not on the representation. Our major resu... |

12 |
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(Show Context)
Citation Context ...examples of black box groups are matrix groups and permutation groups. This effort was initially motivated by recent efforts to identify the structure of matrix groups defined over finite fields (see =-=[3]-=- [7] and [9] in this volume). However, an important advantage of working in a black box setting is that the techniques rely only on the structure of the group and not on the representation. Our major ... |

11 |
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(Show Context)
Citation Context ...black box groups are matrix groups and permutation groups. This effort was initially motivated by recent efforts to identify the structure of matrix groups defined over finite fields (see [3] [7] and =-=[9]-=- in this volume). However, an important advantage of working in a black box setting is that the techniques rely only on the structure of the group and not on the representation. Our major result is th... |

1 |
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(Show Context)
Citation Context ...lementations at this point. In this situation, heuristic methods for generating nearly uniformly distributed group random elements are used. A novel heuristic for such random elements is described in =-=[4]-=- and has been used by several authors (see [9] in this volume). It is also required that we know the prime factorization for each divisor of |G| of the form 2i −1, 1 ≤ i ≤ n. This allows us to use the... |