Volume 17, Issue 3

Abstract. This is the second in a series of papers investigating the space of Brauer relations of a finite group, the kernel of the natural map from its Burnside ring to the rational representation ring. The first paper classified all primitive Brauer relations, that is those that do not come from proper subquotients. In the case of quasi-elementary groups the description is intricate, and it does not specify groups that have primitive relations in terms of generators and relations. In this paper we provide such a classification in terms of generators and relations for quasi-elementary groups of order p a q .

Abstract. We study symplectic representations of groups of the form (A⋊B)⋊C${(A\rtimes B)\rtimes C}$, where A , B are finite cyclic groups of coprime orders, and C is an infinite cyclic group.

Abstract. Let G be a finite group. A subgroup H of G is called pronormal in G if for each g∈G${g\in G}$, the subgroups H and H g are conjugate in 〈H,Hg〉${\langle H, H^g \rangle }$. A subgroup H of G is called weakly pronormal in G if there exists a subgroup K of G such that G=HK${G=H\,K}$ and H∩K${H\cap K}$ is pronormal in G . The main purpose of this paper is to investigate the structure of G under the assumption that some subgroups of G are weakly pronormal in G .

Abstract. In this paper we prove that a unipotent automorphism group H acting upon a soluble group with finite Prüfer rank is nilpotent. Furthermore, if H is finitely generated and the cardinality of a minimal set of generators is t , it is then possible to find an upper bound for the nilpotency class of H which depends only on n and t .

Abstract. We settle an old conjecture of Karrass and Solitar by proving that a finitely generated subgroup of a non-trivial free product G=A*B${G = A\ast B}$ has finite index if and only if it intersects each non-trivial normal subgroup of G non-trivially. This holds, more generally, for subgroups of finite Kurosh rank.

Abstract. We give an infinite family of torsion-free groups that do not satisfy the unique product property. For these examples, we also show that each group contains arbitrarily large sets whose square has no uniquely represented element.

Abstract. We describe a simple scheme for constructing finitely generated monoids in which left-divisibility is a linear ordering, and for practically investigating these monoids. The approach is based on subword reversing, a general method of combinatorial group theory, and connected with Garside theory, here in a non-Noetherian context. As an application we describe several families of ordered groups whose space of left-invariant orderings has an isolated point, including torus knot groups and some of their amalgamated products.