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...

The Burnside problem about periodic groups asks whether any finitely generated group with the law x ≡ 1 is necessarily finite. This is proven only for n ≤ 4 and n = 6. A negative solution to the Burnside problem for odd n ≫ 1 was given by Novikov and Adian. The article presents a discussion of a recent solution of the Burnside problem for even exponents n ≫ 1 and related results. 1991 Mathemati...

Witt#1: The Burnside Theorem [completed, not proofread] Theorem 1, the Burnside theorem ([1], 19.10). Let G be a finite group, and let X and Y be finite G-sets. Then, the following two assertions A and B are equivalent: Assertion A: We have X ∼ = Y , where ∼ = means isomorphism of G-sets.

In this paper, we construct a q-deformation of the Witt-Burnside ring of a profinite group over a commutative ring, where q ranges over the ring of integers. When q = 1, this coincides with the Witt-Burnside ring introduced by A. Dress and C. Siebeneicher (Adv. Math. 70 (1988), 87-132). To achieve our goal we first show that there exists a q-deformation of the necklace ring of a profinite group...