On the groups generated by two operators

Miller, G. A.

doi:10.1090/s0002-9904-1901-00826-9

Cited by 65 publications

(56 citation statements)

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“…Indeed it had been shown by Liebeck and Shalev [14, Proposition 6.2] that PSp 4 (q) is not (2, 3)-generated for p = 2, 3. And SL 4 (2) ∼ = Alt (8) is not (2, 3)-generated by a result of Miller [17].…”

Section: Proofs Of Negative Resultsmentioning

confidence: 98%

Uniform (2,k)-generation of the 4-dimensional classical groups

2012

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Section: Proofs Of Negative Resultsmentioning

confidence: 98%

Uniform (2,k)-generation of the 4-dimensional classical groups

2012

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“…For the alternating groups, this was first stated in a 1901 paper of G.A. Miller [47]. In 1962 it was extended by Steinberg [54] to the simple groups of Lie type, and post-Classification, Aschbacher and Guralnick [2] completed the proof by analysing the remaining sporadic groups.…”

Section: Introductionmentioning

confidence: 95%

Generation and random generation: From simple groups to maximal subgroups

Burness¹,

Liebeck

Shalev

2013

Advances in Mathematics

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Abstract. Let G be a finite group and let d(G) be the minimal number of generators for G. It is well known that d(G) = 2 for all (non-abelian) finite simple groups. We prove that d(H) ≤ 4 for any maximal subgroup H of a finite simple group, and that this bound is best possible.We also investigate the random generation of maximal subgroups of simple and almost simple groups. By applying a recent theorem of Jaikin-Zapirain and Pyber we show that the expected number of random elements generating such a subgroup is bounded by an absolute constant.

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“…A knowledge of pairs of generating elements for exceptional groups such as in Theorem 2 goes back into the history. As early as 1901, Miller [13] found pairs of generating elements, one of which is an involution, for alternating groups; in 1959, Albert and Thompson [14] specified such pairs for A~(q). For linear groups of degrees 2 and 3, Levchyuk [15,16] described all (up to equality) subgroups with a given diagonal matrix.…”

Section: Some Consequencesmentioning

confidence: 99%

Generating triples of involutions for lie-type groups over a finite field of odd characteristic. II

Nuzhin¹

1997

Algebr Logic

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We describe simple Lie-type groups of rank l ~_ 3 over a finite field of odd characteristic, generated by three involutions of which two are commuting. Previously, the answers to the similar questior#s were given for alternating groups, for Lie-type groups over a finite field of characteristic 2, and for Lie-type groups of rank l >_ 4 over a finite field of odd characteristic. These furnish a description of finite simple non-Abelian groups, distinct from 26 sporadic groups, generated by three involutions two of which are commuting, thus giving a partial answer to question 7.30

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On the groups generated by two operators

Cited by 65 publications

References 0 publications

Uniform (2,k)-generation of the 4-dimensional classical groups

Uniform (2,k)-generation of the 4-dimensional classical groups

Generation and random generation: From simple groups to maximal subgroups

Generating triples of involutions for lie-type groups over a finite field of odd characteristic. II

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