## A Non-Constructive Recognition Algorithm for the Special Linear and Other Classical Groups (0)

Venue: | In Groups and Computation II |

Citations: | 15 - 1 self |

### BibTeX

@INPROCEEDINGS{Celler_anon-constructive,

author = {Frank Celler and C.R. Leedham-Green},

title = {A Non-Constructive Recognition Algorithm for the Special Linear and Other Classical Groups},

booktitle = {In Groups and Computation II},

year = {},

pages = {61--67},

publisher = {American Mathematical Society}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In the rst part of this note we present a Monte Carlo algorithm that decides if a given set of matrices generates a group containing the special linear group. In the second part we give timings and extend the algorithm to the other classical groups. 1. Introduction Inspired by the Neumann-Praeger algorithm [6] for recognising whether the subgroup of GL(d; q) generated by some nite subset contains SL(d; q), we present our own version. The two methods are both Monte Carlo algorithms, in that they will either give a positive answer with proof, or a negative answer to a prescribed degree of condence, subject to the fact that both require a number of random elements of the group, which will in fact be produced by multiplying elements of the given generating set X. Both algorithms use the Aschbacher classication of subgroups of classical groups, see [3], and hence will extend to recognising whether subsets of other classical groups generate the whole group. Our algorithm is easier to ...

### Citations

98 |
The subgroup structure of the finite classical groups
- Kleidman, Liebeck
- 1990
(Show Context)
Citation Context ...dom elements of the group, which will in fact be produced by multiplying elements of the given generating set X. Both algorithms use the Aschbacher classication of subgroups of classical groups, see [=-=3]-=-, and hence will extend to recognising whether subsets of other classical groups generate the whole group. Our algorithm is easier to generalise to other classical groups, simply because it is by far ... |

51 |
The computer calculation of modular characters (the meataxe).â€™ â€˜Computational group theory
- Parker
- 1982
(Show Context)
Citation Context ...lassical group modulo scalars. Having found an element with nullity one we use Norton's irreducibility test to decide whether the group generated by the g 1 g acts indeed absolutely irreducibly, see [=-=7-=-]. If it does, we look for an isomorphism between V and V . It is easy to check if such an isomorphism gives a symplectic or orthogonal formsxed by G modulo scalars. The unitary cases can be handled ... |

39 | Praeger: A recognition algorithm for the special linear groups, manuscript
- Neumann, Cheryl
- 1990
(Show Context)
Citation Context ...rates a group containing the special linear group. In the second part we give timings and extend the algorithm to the other classical groups. 1. Introduction Inspired by the Neumann-Praeger algorithm =-=[6]-=- for recognising whether the subgroup of GL(d; q) generated by somesnite subset contains SL(d; q), we present our own version. The two methods are both Monte Carlo algorithms, in that they will either... |

31 |
The Geometry of the Classical Groups. Heldermann Verlag Berlin
- Taylor
- 1992
(Show Context)
Citation Context ... group in characteristic two, we need also to determine the image of the generators under the spinor norm. This we do by using the discriminants of the corresponding Wall forms of the generators, see =-=[9]-=- page 163. The type of the orthogonal group (O + or O ) can be checked by choosing a basis consisting of hyperbolic pairs fe i ; f i g for the symmetric form and examining the quadratic form restricte... |

18 | Calculating the order of an invertible matrix
- Celler, Leedham-Green
- 1997
(Show Context)
Citation Context ...erator for G and an implementation of an algorithm that hopes to determine the order of an element of GL(d; q), modulo scalars, in the time that it takes to perform a few multiplications. See [2] and =-=[1]-=- for details. Remark. The order algorithm can fail to give a precise answer in case of large orders, because of the problems of factorising large integers, but it always gives enough information to de... |

6 |
Generating random elements of a group
- Celler, Leedham-Green, et al.
- 1995
(Show Context)
Citation Context ...deed the case, by examining at most six random elements of G, for any values of d and q. We assume throughout that we can construct random elements of G. In practice we use the algorithm described in =-=[2]-=-. After a pre-processing, this algorithm returns pseudo-random elements of G, each requiring a single matrix multiplication. As the estimate of six elements is purely experimental, a more rigorous app... |

2 |
On the minimal degrees of projective representations of the Chevalley groups
- Landazuri, Seitz
- 1974
(Show Context)
Citation Context ...y [5], and of course their order must divide that of GL(d; q). If they are Chevalley groups dened in a dierent characteristic from that of q, then the lower bounds of Landazuri/Seitz/Zalesskii, see [4=-=]-=- and [8], for non-trivial representations of Chevalley groups in unnatural characteristics reduce the number of cases to be considered to an easily manageable set. 4 FRANK CELLER AND C.R.LEEDHAM-GREEN... |

1 |
On the orders of maximal subgroups of the classical groups
- Liebeck
- 1985
(Show Context)
Citation Context ...oes not divide r 2 exp(Sp(m; r)): 9. Central extensions of almost simple groups. The almost simple groups that could arise, and are not amongst those already described, have order less than q 3d by [=-=-=-5], and of course their order must divide that of GL(d; q). If they are Chevalley groups dened in a dierent characteristic from that of q, then the lower bounds of Landazuri/Seitz/Zalesskii, see [4] a... |