Arithmetic results on orbits of linear groups

A linear group G ≤ GL(V ), where V is a finite vector space, is called 1 2transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the 1 2 -transitive linear groups. As a consequence we complete the determination of the finite 3 2 -transitive permutation groupsthe transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the (k + 1 2 )-transitive groups for integers k ≥ 2., where R = SL 2 (5) and Z 0 is a subgroup of F * 13 2 of order 28, and G ∩ GL 2 (13 2 ) = Z 0 R has orbits on 1-spaces of sizes 20, 30, 60, 60.Combining Theorem 1 with the soluble case in [13,14] and the p-modular case in [8, Theorem 6], we have the following classification of 1 2 -transitive linear groups. In the statement, for q an odd prime power, S 0 (q) is the subgroup of GL 2 (q) of order 4(q − 1) consisting of all monomial matrices of determinant ±1. Proof of Theorem 1Throughout the proof, we shall use the following well-known result about the structure of Frobenius complements, due to Zassenhaus. Proposition 2.1 ([15, Theorem 18.6]) Let G be a Frobenius complement. (i) The Sylow subgroups of G are cyclic or generalized quaternion.(ii) If G is insoluble, then it has a subgroup of index 1 or 2 of the form SL 2 (5)×Z,where Z is a group of order coprime to 30, all of whose Sylow subgroups are cyclic.The following result is important in our inductive proof of Theorem 1.Proposition 2.2 Let R = SL 2 (5), let p > 5 be a prime, and let V be a nontrivial absolutely irreducible F q R-module, where q = p a . Regard R as a subgroup of GL(V ), and let G be a group such that R ⊳ G ≤ ΓL(V ).(i) If R is semiregular on V ♯ , then dim V = 2.

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“…Hence N is faithful on U ; it is also absolutely irreducible, as in the proof of [8,Lemma 12.2]. Hence (i) holds, and (ii) follows.…”

Section: By [14 Theorem 11] G Acts Primitively As a Linear Group Onmentioning

confidence: 62%

The classification of \frac{3}2-transitive permutation groups and \frac{1}2-transitive linear groups

Liebeck

Praeger

Saxl

2019

Proc. Amer. Math. Soc.

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“…Since N = F * (H) is the central product of F (H) and the layer E(H), where E(H)/Z is the direct product of the minimal non-Abelian normal subgroups of H/Z it follows that N/Z = Soc(H/Z) as claimed. By [5,Lemma 12…”

Section: Proof Of Theorem 12mentioning

confidence: 97%

“…there is no coprime subgroup L ≤ GL(V ) strictly containing G. In the following, we give a structure theorem of such groups very similar to a result about maximal solvable primitive linear group (see [21,Lemma 2.2] and [23, § §19-20]). Our proof uses ideas similar to those can be found in [5], [8], and [23]. For the convienience of the reader, we give a self-contained proof here.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Random bases for coprime linear groups

Duyan¹,

Halasi

Podoski

2019

Journal of Group Theory

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“…Let

R

be a ring and let

V

be an

R

‐module with

| V | < \infty

. Given a linear group

G ⩽ GL (V)

, it is a classical problem (for

R = {boldF}_{q}

) to relate arithmetic properties of the number of orbits of

G

on its natural module

V

to geometric and group‐theoretic properties of

G

; see, for example, and the references therein. This problem is closely related to the enumeration of irreducible characters (and hence of conjugacy classes).…”

Section: Introductionmentioning

confidence: 99%

The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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Let frakturO be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over frakturO, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of frakturO. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐p groups.

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Arithmetic results on orbits of linear groups

Cited by 19 publications

References 48 publications

The classification of \frac{3}2-transitive permutation groups and \frac{1}2-transitive linear groups

The classification of \frac{3}2-transitive permutation groups and \frac{1}2-transitive linear groups

Random bases for coprime linear groups

The average size of the kernel of a matrix and orbits of linear groups

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