Abstract

Given a finite group G and a field F, a G-set X gives rise to an F[G]-permutation module F[X]. This defines a map from the Burnside ring of G to its representation ring over F. It is an old problem in representation theory, with wide-ranging applications in algebra, number theory, and geometry, to give explicit generators of the kernel K_F(G) of this map, i.e. to classify pairs of G-sets X, Y such that F[X] is isomorphic to F[Y]. When F has characteristic 0, a complete description of K_F(G) is now known. In this paper, we give a similar description of K_F(G) when F is a field of characteristic p>0 in all but the most complicated case, which is when G has a subquotient that is a non-p-hypo-elementary (p,p)-Dress group.

Highlights

  • We study which finite G-sets X, Y, for a finite group G, give rise to isomorphic linear permutation representations over a field of positive characteristic

  • Since every finite G-set is a finite disjoint union of transitive G-sets, and every transitive G-set is isomorphic to a set of the form G/H, where H is a subgroup of G, with G/H isomorphic to G/H′ if and only if H is G-conjugate to H′, we deduce that as a group B(G) is free abelian on the set of conjugacy classes of subgroups of G

  • We will sometimes refer to these conjugacy classes of subgroups as the terms of Θ, so that if Θ ∈ B(G) and H is a subgroup of G, we may talk about the coefficient of H in Θ

Read more

Summary

Introduction

We study which finite G-sets X, Y , for a finite group G, give rise to isomorphic linear permutation representations over a field of positive characteristic. If G itself is not a (p, q′)-Dress group for any prime number q′, PrimF (G) is cyclic, and is generated by any relation of the form Θ = G + H G aH H, where aH ∈ Z This almost immediately implies the conclusions of the theorem when G is not soluble – see part (iii) of the conclusion. To prove part (B) of the theorem, it remains to exhibit explicit relations of the form Θ = G + H G aH H for groups G appearing in the theorem that are not (p, q)-Dress for any prime number q, and to separately deal with (p, q)-Dress groups that do not have a non-trivial normal p-subgroup, i.e. that are q-quasi-elementary.

Basic properties and induction theorems
Main reduction in soluble groups
Explicit relations
Full Text
Paper version not known

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.