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

Abstract

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.

The 2-closure of a 32-transitive group in polynomial time

“…As already said, the classification of 3 2 -transitive permutation groups was recently completed [12]. Below we summarize the main results from [12] for groups of sufficiently large degree.…”

“…Later Passman classified solvable 3 2 -transitive groups [16,17,19]. Almost simple 3 2 -transitive groups were described in [3], and the final step towards the classification of 3 2 -transitive groups was done recently in [11,12]. The purpose of this paper is to show that this classification allows to prove that the 2-closure problem for 3 2 -transitive groups and the isomorphism problem for 3 2 -homogeneous coherent configurations associated with these groups can be solved in polynomial time in degree of a group.…”

The symmetry property of (n,k)‐arrangement graph

“…The 2‐transitive groups were first listed by Cameron (see [6, Tables 7.3 and 7.4] for example), and we may pick up the 3‐transitive groups (also see [29, section 3]).…”

“…The sharply ‐transitive groups were classified by Jordan for and by Zassenhaus for or 3; see [15, Section 7.6]. For more information about sharply ‐transitive groups, we also refer to [37, Lemma 3.4] and [29, Section 3].…”

A characterization of exceptional pseudocyclic association schemes by multidimensional intersection numbers

“…transitive permutation groups shows that in most cases the coherent configuration of a -transitive group is pseudocyclic, see Subsection 6.1. We cite a part of this classification in the following theorem, see[10, Corollaries 2,3]. Let G be a3 2 -transitive permutation group of degree n. Assume that neither G is 2-transitive or Frobenius nor G ≤ AΓL(1, q) for some q.…”