## A recognition algorithm for special linear groups (1992)

Venue: | Proc. London Math. Soc |

Citations: | 39 - 1 self |

### BibTeX

@ARTICLE{Neumann92arecognition,

author = {Peter M. Neumann and Cheryl E. Praeger},

title = {A recognition algorithm for special linear groups},

journal = {Proc. London Math. Soc},

year = {1992},

volume = {3},

pages = {555--603}

}

### Years of Citing Articles

### OpenURL

### Abstract

Neubiiser asked for an efficient algorithm to decide whether the subgroup of the general linear group GL(d, q) generated by a given set X of non-singular d X d matrices over a finite field ¥q contains the special linear group SL(rf, q).

### Citations

385 |
Atlas of Finite Groups
- Conway
- 1985
(Show Context)
Citation Context ...2 3 .3.5.7.11.19, we would find that the only set of numbers satisfying our constraints are d = 10, r = 11, s = 19, whereas 7T has no irreducible representation of degree 10 over any field (see Atlas =-=[3]-=- or Parker [23]). Therefore T cannot be Jx either, and we are left with the Mathieu groups. Suppose now that T is Af23 or M24. Since d 2= 11 the constraints on d, r, s are satisfied only by d = 11, r ... |

310 | The Art of Computer Programming. Vol. 2. Seminumerical Algorithms - Knuth - 1997 |

264 |
Endliche Gruppen I
- Huppert
- 1967
(Show Context)
Citation Context ...a multiple of d — 1, where $ is the Euler phi function. Lemma 2.2 follows easily from Lemma 2.1. And Lemma 2.1 is quite well known. Since most of it is to be found as Satz II.7.3 on p. 187 of Huppert =-=[11]-=-, we leave the details to the reader. LEMMA 2.3. Suppose that d 2* 3 and Sh(d, q)^G =sGL(d, q). The proportion of elements of G which are irreducible is at least l/(d + 1), and the proportion of eleme... |

123 |
Linear groups with an exposition to the Galois field theory
- Dickson
- 1901
(Show Context)
Citation Context ...lars over subfields: G is conjugate to a subgroup of GL(d, p c ). Z for some divisor c of b. If such a group is to contain suitable elements x, y then a must be coprime with d(d — 1), where a := b/c. =-=(5)-=- Nearly simple groups: there exists G(> «a G such that GJZ is a non-abelian simple group T (which is very much smaller than PSL(d, q)) and GIZ =£ Aut T; furthermore, Go is primitive and absolutely irr... |

100 |
On the maximal subgroups of the finite classical groups
- Aschbacher
- 1984
(Show Context)
Citation Context ...scalars, y has order 2 or 4 modulo scalars, and G/R is isomorphic to a subgroup of the affine group ASL(2, 3). If d = 2 then q is odd and G/Z is Alt(4) or Sym(4). We shall use a theorem of Aschbacher =-=[1]-=- to prove THEOREM 1. Let G be a subgroup of the general linear group GL(rf, q) that contains an irreducible element x and a nearly irreducible element y. Then G is of one of the types listed in Catalo... |

37 |
Finite Groups III
- Huppert, Blackburn
- 1982
(Show Context)
Citation Context ...even then in fact n"° *£ d ^ aoi ^ aon. We may assume that n 2 s 3 and it follows that a0 =s 2. If a0 = 2 then a = 2c, b = boc where 60 is odd, and there is only one possibility for (n, d, i), namely =-=(3,12, 3)-=-. In this case the prime s, which is a primitive prime divisor of q u — 1, would also have to divide (u 6 + l)(w 4 — I)(u 2 + 1) and this is clearly not possible. Thus we must have a0 = 1, b = ac for ... |

28 | On the orders of maximal subgroups of the finite classical groups - Liebeck - 1985 |

26 | The maximal factorizations of the finite simple groups and their automorphism groups - Liebeck, Praeger, et al. - 1990 |

26 |
Determination of the ordinary and modular ternary linear groups. Trans. AMS 12
- Mitchell
- 1911
(Show Context)
Citation Context ...t(6), Alt(7) or PSL(2, 7). Most of this, all except the explicit list of simple groups in Part (7) in fact, comes directly from Aschbacher's Theorem. But again, it goes back much earlier, to Mitchell =-=[19]-=- and to Hartley [9]. The nearly simple groups may be analysed further as follows. The group Alt(5) occurs (as G/Z where G is irreducible) if and only if 5 divides |GL(3, q)\, that is, if and only ifp ... |

25 |
On an algorithm for finding a base and strong generating set for a group given by generating permutations
- Leon
- 1980
(Show Context)
Citation Context ...are 2047 of them). The remaining 759 non-zero vectors will then form a single G-orbit and perhaps the easiest way to distinguish between M23 and A/24 is to apply the methods of Sims (see [24] or Leon =-=[14]-=-) to this permutation representation of degree 759 in order to find \G\. We leave details to the reader. PROCEDURE BIG MATHIEU. Given G *£ GL(11, 2) such that G is not contained in a conjugate of FL(l... |

24 |
Transitive linear groups and linear groups which contain irreducible subgroups of prime order
- Hering
- 1985
(Show Context)
Citation Context ...efore that / is odd and that n<d = 2i^2n. Then, usings574 PETER M. NEUMANN AND CHERYL E. PRAEGER [15, Theorem 1.1] again, we see that n^5, and the possibilities for (n, d, i) are (3,6,3), (4,6,3) and =-=(5,10,5)-=-. If d = 6 then q has order 5 modulo s and this is inconsistent with the facts that q is a power of u and that s divides u 2 ' -1 for some y =s n ^ 4. If {n, d) = (5, 10) then s divides q 2j - 1 for s... |

13 |
Determination of the ternary collineation groups whose coefficients lie
- Hartley
- 1926
(Show Context)
Citation Context ...2, 7). Most of this, all except the explicit list of simple groups in Part (7) in fact, comes directly from Aschbacher's Theorem. But again, it goes back much earlier, to Mitchell [19] and to Hartley =-=[9]-=-. The nearly simple groups may be analysed further as follows. The group Alt(5) occurs (as G/Z where G is irreducible) if and only if 5 divides |GL(3, q)\, that is, if and only ifp = 5or<? = ±l (mod 5... |

5 |
Some subgroups of SLn (F2
- McLaughlin
- 1969
(Show Context)
Citation Context ... (of degree at least 2). If jy is such an element then y qd ~ 2 ~ x is a transvection and the irreducible subgroups of GL,(d, 2) that are generated by transvections have been classified by McLaughlin =-=[17]-=-. This allows one to design a neat recognition algorithm in this case. We give no details partly because one of the problems it gives rise to is the recognition of the symplectic and orthogonal groups... |

2 |
RF. Screening surgeons for human immunodeficiency virus (HIV). A cost-effectiveness analysis. Ann Intern Med
- unknown authors
- 1987
(Show Context)
Citation Context ...l theorem in a series of papers the first of which has appeared as [28]. Another more general study has been published (as was brought to our attention after this work was complete) also by Dempwolff =-=[4]-=-, whose methods and applications are, however, completely different from ours. The list is as follows. CATALOGUE 2. List of nearly simple subgroups G ofGL(d, q), where d 2= 4, that contain a primitive... |

2 |
An algorithm for determining the simplicity of a modular group representation
- Michler
- 1988
(Show Context)
Citation Context ...se null-space is small enough to allow an exhaustive search for vectors generating proper submodules). This element of uncertainty may, in principle, be removed using an algorithm proposed by Michler =-=[18]-=-. In practice, however, his method seems significantly more costly than the sort of calculations that we have proposed hitherto. NOTE 11.2. As we have already pointed out, if SL(d,q)^G then the probab... |

1 |
Fast recognition of alternating and symmetric groups
- CAMERON, CANNON
(Show Context)
Citation Context ...ible element x and a nearly irreducible element y. We take d 5= 4 and q = p b where p is a prime number; Z denotes the subgroup of scalar matrices contained in G. (1) Any group G containing SL(d, q). =-=(2)-=- Monomial groups: G is conjugate to a subgroup of the monomial group GL(1, q) wr Sym(d), that is, G preserves a direct-sum decomposition V = Vx ©... © Vd, where dim Vt = l for all / and we may take K,... |

1 |
Uber Gruppen des Grades p oder p
- FROBENIUS
- 1902
(Show Context)
Citation Context ...cible; if it were not contained in Z then Co would be imprimitive as linear group, whereas, since it contains an element of order r, it is certainly primitive. It follows (see, for example, Frobenius =-=[6]-=- or Huppert and Blackburn [12, Chapter XIII]) that GJ(Z n Go) = PSL(2, r), and hence that the group (£/„, Uo) modulo scalars is isomorphic to PSL(2, r). Finally,s584 PETER M. NEUMANN AND CHERYL E. PRA... |

1 |
A central limit theorem on GLn(Fy
- GOH, SCHMUTZ
- 1990
(Show Context)
Citation Context ..., q) is the proportion of prima elements in a group G that contains SL{d, q) then e{d, q)—>l as d—»°° uniformly in q. If this is true, as we believe (for some confirmation of (ii) see Goh and Schmutz =-=[7]-=-), then it would only be necessary to examine a small number of randomly chosen elements of our given group G, assuming that G is known by this stage to be irreducible (see Note 11.1), in order to dec... |

1 |
On the smallest degrees of projective representations of the groups PSL(n, q)\ Canad
- HARRIS, HERING
- 1971
(Show Context)
Citation Context ...t n ^ 3. Considering the smallest possible degrees of faithful linear or projective representations of PSL(n, u) we see that d s= u n ~ x — 1 except if n = 3 and u = 2 or u = 4 (see Harris and Hering =-=[8]-=-). We leave these two exceptions aside for the moment. Since r>d>u ^u(h we certainly have that x is a semisimple element of PSL(n, u), and so r divides u' -1 for some i =s n. Then the inequalities sat... |

1 |
A search for analogues of the Mathieu groups
- PARKER, NIKOLAI
(Show Context)
Citation Context ...ect Classification: 20-04, 20C40, 20C20. Proc. London Math. Soc. (3) 65 (1992) 555-603.s556 PETER M. NEUMANN AND CHERYL E. PRAEGER machine computation by E. T. Parker and Paul J. Nikolai in 1958 (see =-=[21]-=-). Many variations and improvements to the basic idea are possible and a highly efficient modern version has been prepared by Peter Cameron and John Cannon PI- Our approach to the problem for matrix g... |