The purpose of this paper is twofold. First we aim to unify previous work by the first two authors, A. Garsia, and C. Reutenauer (see [2], [3], [4], [5] and [10]) on the structure of the descent algebras of the Coxeter groups of type An and Bn. But we shall also extend these results to the descent algebra of an arbitrary finite Coxeter group W. The descent algebra, introduced by Solomon in [14]...

We show that there is an exact sequence of biset functors over p-groups 0 → Cb j −→B∗ Ψ −→D → 0 where Cb is the biset functor for the group of Borel-Smith functions, B ∗ is the dual of the Burnside ring functor, D is the functor for the subgroup of the Dade group generated by relative syzygies, and the natural transformation Ψ is the transformation recently introduced by the first author in [5]...

For every profinite group G, we construct two covariant functors ∆G and APG from the category of commutative rings with identity to itself, and show that indeed they are equivalent to the functor WG introduced in [A. Dress and C. Siebeneicher, The Burnside ring of profinite groups and the Witt vectors construction, Adv. Math. 70 (1988), 87-132]. We call ∆G the generalized Burnside-Grothendieck ...

We establish some results about large restricted Lie algebras similar to those known in the Group Theory. As an application, we use this group-theoretic approach to produce some examples of restricted as well as ordinary Lie algebras which can serve as counterexamples for various Burnside type questions.

In this paper, we show that integral fusion categories with rational structure constants admit a natural group of symmetries given by the Galois their character tables. Based on these symmetries, generalize well-known result Burnside from representation theory finite groups. More precisely, any row corresponding to non-invertible object in table weakly category contains zero entry.

After we have given a survey on the Burnside ring of a finite group, we discuss and analyze various extensions of this notion to infinite (discrete) groups. The first three are the finite-G-set-version, the inverselimit-version and the covariant Burnside group. The most sophisticated one is the fourth definition as the equivariant zero-th cohomotopy of the classifying space for proper actions. ...

Using the Burnside ring theoretic methods a new setting and a complete description of the Artin exponent A(G) of finite p-groups was obtained in [4]. In this paper, we compute A(G) for any finite group G – hence providing the global version of [4].