Mathematics

Let W be a 2 -dimensional Coxeter group, that is, a one with 1 m st + 1 m sr + 1 m tr ≤1 for all triples of distinct s,t,r∈S . We prove that W is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary W ), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Świątkowski, groups acting properly and cocompactly on buildings of type W are also biautomatic. We also show that the fellow traveller property for the natural language fails for W= A ˜ 3 .

Let F be a class of group and G a finite group. Then a set Σ of subgroups of G is called a \emph{ G -covering subgroup system} for the class F if G?�F whenever Σ?�F . We prove that: {\sl If a set of subgroups Σ of G contains at least one supplement to each maximal subgroup of every Sylow subgroup of G , then Σ is a G -covering subgroup system for the classes of all ? -soluble and all ? -nilpotent groups, and for the class of all ? -soluble P?T -groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook.

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow 2 -subgroup of AGL(2,n), each term of the chain is defined as the normalizer of the previous one in the symmetric group on 2 n letters. Partial results and computational experiments lead us to conjecture that, for large values of n , the index of a normalizer in the consecutive one does not depend on n . Indeed, there is a strong evidence that the sequence of the logarithms of such indices is the one of the partial sums of the numbers of partitions into at least two distinct parts.

In a previous paper we generalized the definition of a multilinear map to arbitrary groups and introduced two multiparty key-exchange protocols using nilpotent groups. In this paper we have a closer look at the protocols and will address some incorrect cryptanalysis which have been proposed.

In this paper we introduce and study some geometric objects associated to Artin monoids. The Deligne complex for an Artin group is a cube complex that was introduced by the second author and Davis (1995) to study the K(\pi,1) conjecture for these groups. Using a notion of Artin monoid cosets, we construct a version of the Deligne complex for Artin monoids. We show that for any Artin monoid this cube complex is contractible. Furthermore, we study the embedding of the monoid Deligne complex into the Deligne complex for the corresponding Artin group. We show that for any Artin group this is a locally isometric embedding. In the case of FC-type Artin groups this result can be strengthened to a globally isometric embedding, and it follows that the monoid Deligne complex is CAT(0) and its image in the Deligne complex is convex. We also consider the Cayley graph of an Artin group, and investigate properties of the subgraph spanned by elements of the Artin monoid. Our final results show that for a finite type Artin group, the monoid Cayley graph embeds isometrically, but not quasi-convexly, into the group Cayley graph.

The groups d V n are an infinite family of groups, first introduced by C. Martínez-Pérez, F. Matucci and B. E. A. Nucinkis, which includes both the Higman-Thompson groups V n (=1 V n ) and the Brin-Thompson groups nV(=n V 2 ) . A description of the groups Aut( G n,r ) (including the groups G n,1 = V n ) has previously been given by C. Bleak, P. Cameron, Y. Maissel, A. Navas, and F. Olukoya. Their description uses the transducer representations of homeomorphisms of Cantor space introduced a paper of R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskii, together with a theorem of M. Rubin. We generalise the transducers of the latter paper and make use of these transducers to give a description of Aut(d V n ) which extends the description of Aut(1 V n ) given in the former paper. We make use of this description to show that Out(d V 2 )≅Out( V 2 )≀ S d , and more generally give a natural embedding of Out(d V n ) into Out( G n,n−1 )≀ S d .

A group G possesses the Magnus property if for every two elements u , v∈G with the same normal closure, u is conjugate to v or v −1 . O. Bogopolski and J. Howie proved independently that the fundamental groups of all closed orientable surfaces possess the Magnus property. The analogous result for closed non-orientable surfaces was proved by O. Bogopolski and K. Sviridov except for one case that was later covered by the author. In this article, we generalize those results, which can be viewed as Magnus extensions for free groups, to a Magnus extension for locally indicable groups and consider the influence of adding a group as a direct factor. For this purpose, we also prove versions of the Freiheitssatz for locally indicable groups and of a result by M. Edjvet adding a group as a direct factor.

S. Gersten announced an algorithm that takes as input two finite sequences K → =( K 1 ,…, K N ) and K → ′ =( K ′ 1 ,…, K ′ N ) of conjugacy classes of finitely generated subgroups of F n and outputs: (1) YES or NO depending on whether or not there is an element θ∈Out( F n ) such that θ( K → )= K → ′ together with one such θ if it exists and (2) a finite presentation for the subgroup of Out( F n ) fixing K → . S. Kalajdžievski published a verification of this algorithm. We present a different algorithm from the point of view of Culler-Vogtmann's Outer space. New results include that the subgroup of Out( F n ) fixing K → is of type VF , an equivariant version of these results, an application, and a unified approach to such questions.

Let G be a finite group, n a positive integer. π(n) denotes the set of all prime divisors of n and π(G)=π(|G|) . The prime graph Γ(G) of G , defined by Grenberg and Kegel, is a graph whose vertex set is π(G) , two vertices p, q in π(G) joined by an edge if and only if G contains an element of order pq . In this article, a new characterization of sporadic simple groups is obtained, that is, if G is a finite group and S a sporadic simple group. Then G≅S if and only if |G|=|S| and Γ(G) is disconnected. This characterization unifies the several characterizations that can conclude the group has non-connected prime graphs, hence several known characterizations of sporadic simple groups become the corollaries of this new characterization.

Cycle sets are known to give non-degenerate unitary solutions of the Yang--Baxter equation and linear cycle sets are enriched versions of these algebraic systems. The paper explores the recently developed cohomology and extension theory for linear cycle sets. We derive a four term exact sequence relating 1-cocycles, second cohomology and certain groups of automorphisms arising from central extensions of linear cycle sets. This is an analogue of a similar exact sequence for group extensions known due to Wells. We also compare the exact sequence for linear cycle sets with that for their underlying abelian groups and discuss generalities on dynamical 2-cocycles.