Let A be an associative algebra over a field k, and let M be a finite family of right A-modules. Study of the noncommutative deformation functor DefM : ar → Sets of the family M leads to the construction of the algebra O(M) of observables and the Generalized Burnside Theorem, due to Laudal [2]. In this paper, we give an overview of aspects of noncommutative deformations closely connected to the...

In this note, we simply (try to) interpret Burnside’s original proof in modern terminology and notation. This is instructive for anyone interested in the evolution of mathematical writing, as Burnside’s language is – at least for the author – essentially unreadable as a rigorous mathematical proof. One can make sense of it, with some effort, and from it construct a proof which is (or at least, ...

Let G be a finite group, and let V and W be finite-dimensional real orthogonal G-modules with V 3 W, and with unit spheres S(V) and S( W) respectively. The purpose of this note is to give necessary sufficient conditions for the existence of a C-map /: S( V) -» S( W) in terms of the Burnside ring of G and its relationship with V and W. Note that if W has a nonzero fixed point, such a G-map alway...

Let G be a finite group and let S be a G-set. The Burnside ring of G has a natural structure of a λ-ring, {λn}n∈N. However, a priori λ (S) can only be computed recursively, by first computing λ(S), . . . , λ(S). In this paper we establish an explicit formula, expressing λ(S) as a linear combination of classes of G-sets.

Some general connections between martingales and character ratios of finite groups are developed. As an application we sharpen the convergence rate in a central limit theorem for the character ratio of a random representation of the symmetric group on transpositions. A generalization of these results is given for Jack measure on partitions. We also give a probabilistic proof of a result of Burn...

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 zero-th equivariant stable cohomotopy of the classifying space for proper ac...

In this paper we investigate the word problem of the free Burnside semigroup satisfying x2 = x3 and having two generators. Elements of this semigroup are classes of equivalent words. A natural way to solve the word problem is to select a unique “canonical” representative for each equivalence class. We prove that overlap-free words and socalled almost overlap-free words (this notion is some gene...