Primitive permutation groups and derangements of prime power order

Burness, Timothy C.; Tong-Viet, Hung P.

doi:10.1007/s00229-015-0795-x

Cited by 13 publications

(9 citation statements)

References 48 publications

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“…Now assume r = 2f + 1. By arguing as in the proof of [9,Lemma 4.6] we deduce that f = 2 m is a 2-power, so r = 2 m+1 + 1 is a Fermat prime and thus m + 1 = 2 l for some l 0.…”

Section: Rank One Groups Of Lie Typementioning

confidence: 72%

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Almost elusive permutation groups

Burness¹,

V.²

2020

Preprint

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Let G be a nontrivial transitive permutation group on a finite set Ω. An element of G is said to be a derangement if it has no fixed points on Ω. From the orbit counting lemma, it follows that G contains a derangement, and in fact G contains a derangement of prime power order by a theorem of Fein, Kantor and Schacher. However, there are groups with no derangements of prime order; these are the so-called elusive groups and they have been widely studied in recent years. Extending this notion, we say that G is almost elusive if it contains a unique conjugacy class of derangements of prime order. In this paper we first prove that every quasiprimitive almost elusive group is either almost simple or 2-transitive of affine type. We then classify all the almost elusive groups that are almost simple and primitive with socle an alternating group, a sporadic group, or a rank one group of Lie type.

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“…Now assume r = 2f + 1. By arguing as in the proof of [9,Lemma 4.6] we deduce that f = 2 m is a 2-power, so r = 2 m+1 + 1 is a Fermat prime and thus m + 1 = 2 l for some l 0.…”

Section: Rank One Groups Of Lie Typementioning

confidence: 72%

“…Proposition 3.6. For k 4, either |Ω| is divisible by distinct primes r, s > k, or (n, k) = (12, 5), (9,4).…”

Section: 1mentioning

confidence: 99%

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Almost elusive permutation groups

Burness¹,

V.²

2020

Preprint

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“…Throughout the section we let n be a positive integer and q = p f such that p is prime and f 1. Our first result is [10,Lemma 2.6].…”

Section: Number Theoretic Preliminariesmentioning

confidence: 89%

Almost elusive classical groups

V.¹

2021

Preprint

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Let G be a transitive permutation group acting on a finite set Ω with |Ω| 2. An element of G is said to be a derangement if it has no fixed points on Ω, and by a theorem of Jordan from 1872, G always contains such an element. In particular by a theorem of Fein, Kantor and Schacher G contains a derangement of prime power order. Nevertheless there exist groups in which there are no derangements of prime order, these groups are called elusive groups. Defining an natural extension of this we say G is almost elusive if it contains a unique conjugacy class of derangements of prime order. In recent work with Burness, we reduced the problem of determining the almost elusive quasiprimitive groups to the almost simple and 2-transitive affine cases. Additionally we classified the primitive almost elusive almost simple groups with socle an alternating group, a sporadic group or a group of Lie type with (twisted) Lie rank equal to 1. In this paper we complete the classification of the primitive almost elusive almost simple classical groups.

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“…By [19,Lemma 2.6], there are no solutions to the equation 2 f − 1 = r ℓ with ℓ 2, so q − 1 must be divisible by at least two distinct primes and thus r(r + 2) q − 1. This implies that r √ q − 1 and one can now check that (1) holds.…”

Section: Almost Simple Groupsmentioning

confidence: 99%

Strongly base-two groups

Burness¹,

Guralnick²

2022

Preprint

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Let G be a finite group, let H be a core-free subgroup and let b(G, H) denote the base size for the action of G on G/H. Let α(G) be the number of conjugacy classes of core-free subgroups H of G with b(G, H) 3. We say that G is a strongly base-two group if α(G) 1, which means that almost every faithful transitive permutation representation of G has base size 2. In this paper we study the strongly base-two finite groups with trivial Frattini subgroup.

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Primitive permutation groups and derangements of prime power order

Cited by 13 publications

References 48 publications

Almost elusive permutation groups

Almost elusive permutation groups

Almost elusive classical groups

Strongly base-two groups

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