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...

We show how automorphisms can be used to reduce signiicantly the resources needed to enforce laws in p-groups. This increases the extent to which Burnside groups with prime-power exponent can be studied in detail. For example, we describe how to construct power-conjugate presentations for the restricted Burnside groups R(5; 4) and R(3; 5) which have orders 2 2728 and 5 2282 respectively. We als...

In this note we present an algorithm for the construction of the unit group of the Burnside ring Ω(G) of a finite group G from a list of representatives of the conjugacy classes of subgroups of G.

We consider the problem of sampling \unlabelled structures", i.e., sampling combi-natorial structures modulo a group of symmetries. The main tool which has been used for this sampling problem is Burnside's lemma. In situations where a signiicant proportion of the structures have no non-trivial symmetries, it is already fairly well understood how to apply this tool. More generally, it is possibl...

We introduce equivariant Burnside groups, new invariants in birational geometry, generalizing symbols groups for actions of finite abelian due to Kontsevich, Pestun, and the second author, study their properties. establish a specialization map type smooth algebraic variety with an action group.

The Tate–Shafarevich set of a group [Formula: see text] defined by Takashi Ono coincides, in the case where is finite, with outer class-preserving automorphisms introduced Burnside. We consider analogs this important group-theoretic object for Lie algebras and associative establish some new structure properties thereof. also discuss open problems eventual generalizations to other algebraic stru...

Abstract Let Q denote the cyclic group of order two. Using Tate diagram we compute RO ( )-graded coefficients Eilenberg–MacLane -spectra and describe their structure as modules over spectrum Burnside Mackey functor. If underlying functor is a Green functor, also obtain multiplicative on coefficients.

Let G be a finite almost simple group with socle G0. A (nontrivial) factorization of is an expression the form G=HK, where factors H and K are core-free subgroups. There extensive literature on factorizations groups, important applications in permutation theory algebraic graph theory. In recent paper, Xia second-named author describe groups solvable factor H. Several infinite families arise con...