Burnside Foster (1861–1917)

Burnside Orders, Burnside Algebras and Partition Lattices

The permutation representation theory of groups has been extended, through quasigroups, to one-sided left (or right) quasigroups. The current paper establishes a link with the theory of ordered sets, introducing the concept of a Burnside order that generalizes the poset of conjugacy classes of subgroups of a finite group. Use of the Burnside order leads to a simplification in the proof of key p...

Burnside rings

1 Let G be a finite group. The Burnside ring B(G) of the group G is one of the fundamental representation rings of G, namely the ring of permutation representations. It is in many ways the universal object to consider when looking at the category of G-sets. It can be viewed as an analogue of the ring Z of integers for this category. It can be studied from different points of view. First B(G) is...

Reading Burnside

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, ...

Free Burnside Semigroups

The Restricted Burnside Problem

In 1902 William Burnside [5] wrote 'A still undecided point in the theory of discontinuous groups is whether the order of a group may be not finite, while the order of every operation it contains is finite'. In modern terminology the most general form of the problem is 'can a finitely generated group be infinite while every element in the group has finite order?'. This question was answered in ...