In this work we investigate the hardness of a computational problem introduced in the recent work of Baumslag et al. [BFNSS11]. In particular, we study the Bn-LHN problem, which is a generalized version of the learning with errors (LWE) problem, instantiated with a particular family of non-abelian groups (free Burnside groups of exponent 3). In our main result, we demonstrate a random self-redu...

In this work we investigate the hardness of a computational problem introduced in the recent work of Baumslag et al. in [5, 6]. In particular, we study the Bn-LHN problem, which is a generalized version of the learning with errors (LWE) problem, instantiated with a particular family of non-abelian groups (free Burnside groups of exponent 3). In our main result, we demonstrate a random self-redu...

Abstract We study the free metabelian group $M(2,n)$ of prime power exponent n on two generators by means invariants $M(2,n)'\to \mathbb {Z}_n$ that we construct from colorings squares in integer grid $\mathbb {R} \times {Z} \cup {R}$ . In particular, improve bounds found Newman for order $M(2,2^k)$ identities , which give information about Burnside $B(2,n)$ and restricted $R(2,n)$

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. To achieve our goal we show that there exists a q-deformation of the necklace ring of a profinite group over a commutative ring. As in the classical case, i.e., the cas...

A well-known property of irreducible characters of finite groups is the fact, proved by Burnside, that if G is a finite group and % : G −→ GL(E) is an irreducible complex representation of dimension dim(%) > 2, then there exists some g ∈ G such that χ%(g) = Tr %(g) is zero. A common argument to prove this relies on the following result (also due to Burnside) on representations of finite cyclic ...

The following basic results on infinite locally finite subgroups of a free m-generŽ . 48 ator Burnside group B m, n of even exponent n, where m ) 1 and n G 2 , n is divisible by 29, are obtained: A clear complete description of all infinite groups that Ž . Ž . are embeddable in B m, n as maximal locally finite subgroups is given. Any Ž . infinite locally finite subgroup L of B m, n is contained...

Yasutaka Nakanishi asked in 1981 whether a 3–move is an unknotting operation. In Kirby’s problem list, this question is called The Montesinos–Nakanishi 3– move conjecture. We define the nth Burnside group of a link and use the 3rd Burnside group to answer Nakanishi’s question; ie, we show that some links cannot be reduced to trivial links by 3–moves. AMS Classification numbers Primary: 57M27 Se...

We define the cohomological Burnside ring B(G,M) of a finite group G with coefficients in a ZG-module M as the Grothendieck ring of the isomorphism classes of pairs [X, u] where X is a G-set and u is a cohomology class in a cohomology group H X(G,M). The cohomology groups H ∗ X(G,M) are defined in such a way that H∗ X(G, M) ∼= ⊕iH∗(Hi,M) when X is the disjoint union of transitive G-sets G/Hi. I...