Some generalized characters associated to a transitive permutation group

Group characters, permutation actions and sharpness

We extend the work which has appeared in papers on sharp characters and originated with Blichfeldt and Maillet to the Burnside ring of a finite group G. We show that the polynomial whose zeros are the set of marks of non-identity subgroups on a faithful G-set X evaluated at X is an integral multiple of the regular G-set, and deduce a result about the size of a base of X . Further consequences f...

On Partitioning the Orbitals of a Transitive Permutation Group

Let G be a permutation group on a set Ω with a transitive normal subgroup M . Then G acts on the set Orbl(M,Ω) of nontrivial M -orbitals in the natural way, and here we are interested in the case where Orbl(M,Ω) has a partition P such that G acts transitively on P. The problem of characterising such tuples (M,G,Ω,P), called TODs, arises naturally in permutation group theory, and also occurs in ...

Incidence Matrices, Permutation Characters, and the Minimal Genus of a Permutation Group

Corollary 1.2 Let G be an almost simple classical group with natural module V of dimension n. If G is linear, assume that G does not contain a graph automorphism. Let 2 ≤ k < n − 1, and let K be the stabilizer of a nondegenerate or totally singular k-space of V . Let P be the stabilizer of a singular 1-space of V . Then the permutation module 1P is a submodule of 1 G K unless one of the followi...

On Transitive Permutation Groups

We assign names and new generators to the transitive groups of degree up to 15, reflecting their structure.

Generating Transitive Permutation Groups