Maximal subgroups of G2(2n)

Cooperstein, Bruce N.

doi:10.1016/0021-8693(81)90241-6

Cited by 140 publications

(171 citation statements)

References 12 publications

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“…Then one may swiftly check in, for example, [5, p xvi ] that |Out(S)| 6 7 log p |S| p , where |G| p denotes the p-part of |G|.…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…L n (q), (n, q) ∈ {(5, 2), (5, 3), (5,4), (6,2), (6,3), (7,2), (8,2), (9, 2)}, U n (q), (n, q) ∈ {(7, 2), (7,3), (7,4), (7,5), (7,7), (7,8), (7,9), (8, 2)}, S n (q), (n, q) ∈ {(8, 2), (8, 3), (10, 2)}, O • n (q), (n, q) ∈ {(9, 3), (9, 5)}, O − n (q), (n, q) ∈ {(10, 2), (10, 3)}, O + n (q), (n, q) ∈ {(10, 2)}. These groups are all of sufficiently small dimension to be covered by the BrayHolt-Roney-Dougal tables.…”

Section: Classical Groupsmentioning

confidence: 99%

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The probability of generating a finite simple group

2013

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We study the probability of generating a finite simple group, together with its generalisation P G,soc G (d), the conditional probability of generating an almost simple finite group G by d elements, given that these elements generate G/ soc G. We prove that P G,soc G (2) 53/90, with equality if and only if G is A 6 or S 6 , and establish a similar result for P G,soc G (3). Positive answers to longstanding questions of Wiegold on direct products, and of Mel nikov on profinite groups, as well as to a conjecture of Holt and Stather, follow easily from our results.

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“…Then one may swiftly check in, for example, [5, p xvi ] that |Out(S)| 6 7 log p |S| p , where |G| p denotes the p-part of |G|.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Section: Classical Groupsmentioning

confidence: 99%

The probability of generating a finite simple group

2013

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“…In some other cases, the maximal subgroups are completely enumerated; in particular for [24,35], for L † = G 2 (q) [24,13], for L † = 2 F ′ 4 (q) [28,12] and for L † = 3 D 4 (q) [25].…”

Section: Group Theory Results and Notationmentioning

confidence: 99%

Transitive projective planes

Gill

2007

Advg

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Abstract.A long-standing conjecture is that any transitive finite projective plane is Desarguesian. We make a contribution towards a proof of this conjecture by showing that a group acting transitively on the points of a non-Desarguesian projective plane must not contain any components. Background definitions and main resultsWe say that a projective plane is transitive (respectively primitive) if it admits an automorphism group which is transitive (respectively primitive) on points. Kantor [22] has proved that a projective plane P of order x admitting a point-primitive automorphism group G is Desarguesian and G ≥ PSL(3, x), or else x 2 + x + 1 is a prime and G is a regular or Frobenius group of order dividing (x 2 + x + 1)(x + 1) or (x 2 + x + 1)x.Kantor's result, which depends upon the Classification of Finite Simple Groups, represents the strongest success in the pursuit of a proof to the conjecture mentioned in the abstract. A corollary of Kantor's result is that a group acts primitively on the points of a projective plane P if and only if it acts primitively on the lines of P. We also know, by a combinatorial argument of Block, that a group acts transitively on the points of a projective plane P if and only if it acts transitively on the lines of P [5].Our primary result is the following:Theorem A. Suppose that G acts transitively on a projective plane P of order x. Then one of the following cases holds:• P is Desarguesian, G ≥ PSL(3, x) and the action is 2-transitive on points;

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“…For the case where G = G 2 , we shall use the fact that the maximal subgroups of G σ = G 2 (q) are known by [14,34].…”

Section: Proof Of Theorem 2 Part (Iii)mentioning

confidence: 99%