On the Probability That a Group Element is p-Singular

Isaacs, I. M.; Kantor, William M.; Spaltenstein, N.

doi:10.1006/jabr.1995.1238

Cited by 33 publications

(28 citation statements)

References 16 publications

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“…We may improve the first assertion of Theorem 5.2 of [IKS95] for finite classical groups using Lemma 2.3 and our discussion in Section 5 on the structure of maximal tori in these groups. Let I be the set of all involutions in a finite classical group G in odd characteristic.…”

Section: Strategy Of Countingmentioning

confidence: 95%

Finding involutions in finite Lie type groups of odd characteristic

Lübeck

Niemeyer²,

Praeger³

2009

Journal of Algebra

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Let G be a finite group of Lie type in odd characteristic defined over a field with q elements. We prove that there is an absolute (and explicit) constant c such that, if G is a classical matrix group of dimension n 2, then at least c/ log(n) of its elements are such that some power is an involution with fixed point subspace of dimension in the interval [n/3, 2n/3). If G is exceptional, or G is classical of small dimension, then, for each conjugacy class C of involutions, we find a very good lower bound for the proportion of elements of G for which some power lies in C.

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Section: Strategy Of Countingmentioning

confidence: 95%

Finding involutions in finite Lie type groups of odd characteristic

Lübeck

Niemeyer²,

Praeger³

2009

Journal of Algebra

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“…The same general approach was applied also in the 1995 paper [4] of Isaacs, Kantor and Spaltenstein proving their stunning theorem bounding below the proportion of p-singular elements in finite permutation groups, and also in the 2005 paper of Cohen and Murray to estimate the proportion of regular semisimple elements in the Lie algebra of a connected reductive algebraic group (see [2,Section 6]) in order to analyse their Las Vegas 'algorithm for Lang's Theorem'.…”

Section: Introductionmentioning

confidence: 94%

Estimating proportions of elements in finite groups of Lie type

Niemeyer

Praeger

2010

Journal of Algebra

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We show that certain closure properties on subsets Q of a finite group G of Lie type enable the ratio |Q |/|G| to be determined by finding the proportions of elements of Q in the maximal tori of G and the proportions of certain related subsets of the Weyl group. We prove fundamental results about these subsets Q , including those necessary for moving between these groups and their automorphism groups, normal subgroups and central quotients. As a sample application of these new techniques, we derive upper and lower bounds for the proportion of elements with order divisible by certain natural sets of primes, for the finite classical groups, thereby improving existing results.

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“…Recall that g ∈ G is p-singular if its order is divisible by p. As Isaacs, Kantor & Spaltenstein [63] and Guralnick & Lübeck [52] show, a group of Lie type in defining characteristic p has a small proportion of p-singular elements. …”

Section: Theorem 81 There Is a Las Vegas Algorithm Which When Givenmentioning

confidence: 99%