Primitive permutation groups with a regular subgroup

This paper starts the classification of the primitive permutation groups (G,Ω) such that G contains a regular subgroup X. We determine all the triples (G,Ω,X) with soc(G) an alternating, or a sporadic or an exceptional group of Lie type. Further, we construct all the examples (G,Ω,X) with G a classical group which are known to us. Our particular interest is in the 8-dimensional orthogonal groups of Witt index 4. We determine all the triples (G,Ω,X) with . In order to obtain all these triples, we also study the almost simple groups G with GPΩ_2n+1(q). The case GU_n(q) is started in this paper and finished in [B. Baumeister, Primitive permutation groups of unitary type with a regular subgroup, Bull. Belg. Math. Soc. 112 (5) (2006) 657–673]. A group X is called a Burnside-group (or short a B-group) if each primitive permutation group which contains a regular subgroup isomorphic to X is necessarily 2-transitive. In the end of the paper we discuss B-groups.     

Primitive permutation groups containing ap-cycle

Project supported in part by National Natural Science Foundation of China and STFG     

Primitive permutation groups of simple diagonal type

LetG be a finite primitive group such that there is only one minimal normal subgroupM inG, thisM is nonabelian and nonsimple, and a maximal normal subgroup ofM is regular. Further, letH be a point stabilizer inG. ThenH∩M is a (nonabelian simple) common complement inM to all the maximal normal subgroups ofM, and there is a natural identification ofM with a direct powerT ^m of a nonabelian simple groupT in whichH∩M becomes the “diagonal” subgroup ofT ^m: this is the origin of the title. It is proved here that two abstractly isomorphic primitive groups of this type are permutationally isomorphic if (and obviously only if) their point stabilizers are abstractly isomorphic. GivenT ^m, consider first the set of all permutational isomorphism classes of those primitive groups of this type whose minimal normal subgroups are abstractly isomorphic toT ^m. Secondly, form the direct productS _m×OutT of the symmetric group of degreem and the outer automorphism group ofT (so OutT=AutT/InnT), and consider the set of the conjugacy classes of those subgroups inS _m×OutT whose projections inS _m are primitive. The second result of the paper is that there is a bijection between these two sets. The third issue discussed concerns the number of distinct permutational isomorphism classes of groups of this type, which can fall into a single abstract isomorphism class.     

Primitive parabolic permutation representations for finite symplectic groups

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite symplectic groups are defined on cosets with respect to maximal parabolic subgroups.     

Primitive parabolic permutation representations of finite special linear and unitary groups

The ranks, degrees, subdegrees, and double stabilizers of permutation representations of finite special linear and unitary groups on cosets of parabolic maximal subgroups are found.     

Primitive parabolic permutation representations for finite simple orthogonal groups in odd dimensions

Ranks, degrees, subdegrees, and double stabilizers of permutation representations for finite simple orthogonal groups in odd dimensions are defined on cosets with respect to maximal parabolic subgroups.     

Collapsing permutation groups

It is shown in [3] that any nonregular quasiprimitive permutation group is collapsing. In this paper we describe a wider class of collapsing permutation groups. Received June 6, 2000; accepted in final form August 11, 2000.     

TheL-sharp permutation groups

An answer is given to a problem proposed by Bannai and Ito for {I, I + s, I + s + t}-sharp permutation group, and the result is used to determineL-sharp groups for L={I, I + 1, I + 3} and {I, I + 2, I + 3}.     

Abelian permutation groups with graphical representations

Journal of Algebraic Combinatorics - In this paper we characterize those automorphism groups of colored graphs and digraphs that are abelian as abstract groups. This is done in terms of basic...     

Covering radius of permutation groups with infinity-norm

The covering radius of permutation group codes are studied in this paper with $l_{\infty }$-metric. We determine the covering radius of the $(p,q)$-type group, which is a direct product of two cyclic transitive groups. We also deduce the maximum covering radius among all the relabelings of this group under conjugation, that is, permutation groups with the same algebraic structure but with relabeled members. Finally, we give a lower bound of the covering radius of the dihedral group code, which differs from the trivial upper bound by a constant of at most one. This improves a result proved by Karni and Schwartz (2018), where the gap between their lower and upper bounds tends to infinity as the code length grows.