J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.

By means of analyzing the notion verbal products groups, we show that soficity, hyperlinearity, amenability, Haagerup property, Kazhdan's property (T) and exactness are preserved under taking $k$-nilpotent while being orderable is not preserved. We also study these properties for solvable Burnside groups. then if two discrete groups sofic, or have their restricted wreath product arising from ni...

In this paper, I describe the structure of the biset functor B sending a p-group P to the group of units of its Burnside ring B(P ). In particular, I show that B is a rational biset functor. It follows that if P is a p-group, the structure of B(P ) can be read from a genetic basis of P : the group B(P ) is an elementary abelian 2-group of rank equal to the number isomorphism classes of rational...

Dear Charlie, I promised to write to you about the "characteristic-free" extension of Richard Brauer's Lemma: if a row permutation and a column permutation have the same effect on a nonsingular matrix, the two permutations must have the same number of cycles of any given length. Differently put: if two permutation matrices are conjugate in the general linear group, they are already conjugate in...

The famous Burnside–Schur theorem states that every primitive finite permutation group containing a regular cyclic subgroup is either 2-transitive or isomorphic to a subgroup of a 1-dimensional affine group of prime degree. It is known that this theorem can be expressed as a statement on Schur rings over a finite cyclic group. Generalizing the latter, Schur rings are introduced over a finite co...

The concept of a permutation representation has recently been extended from groups to quasigroups. Following a suggestion of Walter Taylor, the concept is now further extended to left quasigroups. The paper surveys the current state of the theory, giving new proofs where necessary to cover the general case of left quasigroups. Both the Burnside Lemma and the Burnside algebra appear in this new ...