The group D(P ) of all endo-permutation modules for a finite p-group P is a finitely generated abelian group. We prove that its torsion-free rank is equal to the number of conjugacy classes of non-cyclic subgroups of P . We also obtain partial results on its torsion subgroup. We determine next the structure of Q⊗D(−) viewed as a functor, which turns out to be a simple functor SE,Q , indexed by ...

Morphisms from H to G in the Burnside category of groups are provided by sets with commuting right and left actions of H and G, which are free and finite over G. We extend this construction to groupoids by considering a G-action on a set X as a homomorphism G → Aut(X). This has an alternative expression in terms of “bi-actions,” a variant of the discrete version of a topological construction wh...

Davis, J.F. and G. Katz, Equivariant semicharacteristics and induction, Topology and its Applications 51 (1993) 41-52. The paper defines a semicharacteristic bordism invariant for manifolds equipped with an action of a finite group with specified isotropy. Induction results and the Burnside ring are applied to give homological restrictions on some group actions on odd-dimensional manifolds.

In this paper we compute the Bredon equivariant homology of representation spheres corresponding to the orientable three dimensional representations of cyclic groups and dihedral groups, as well as the symmetric group on three letters equipped with a permutation representation. These computations are greatly simplified by the introduction of a splitting theorem for the Burnside ring Mackey func...

a result of dixon‎, ‎evans and smith shows that if $g$ is a locally (soluble-by-finite) group whose proper subgroups are (finite rank)-by-abelian‎, ‎then $g$ itself has this property‎, ‎i.e‎. ‎the commutator subgroup of $g$ has finite rank‎. ‎it is proved here that if $g$ is a locally (soluble-by-finite) group whose proper subgroups have minimax commutator subgroup‎, ‎then also the commutator s...

We study some properties of the coset poset associated with family subgroups class \(\le 2\) a nilpotent group 3\). prove that under certain assumptions on is simply-connected if and only 2-Engel, 2-connected 2 or less. determine homotopy type for \(4\times 4\) upper unitriangular matrices over \(\mathbb {F}_p\), Burnside groups exponent 3.