Volume 17, Issue 1

Abstract. We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

Abstract. Let F be a finite-rank free group and Z be a finite subset of  F . We give topology-free proofs for two algorithms that yield sub-bases and of  F satisfying that minimize the value . Here, the subgroup is uniquely determined, and Richard Stong showed that a special basis thereof is produced by J. H. C. Whitehead's cut-vertex algorithm. Stong's proof used bi-infinite paths in a Cayley tree and sub-surfaces of a handlebody. We give a new proof that uses edge-cuts of the Cayley tree that are induced by edge-cuts of a Bass–Serre tree. A. Clifford and R. Z. Goldstein used Whitehead's three-manifold techniques to give an algorithm that determines whether or not there exists a basis of F that meets . We replace the topology with the cut-vertex algorithm, and obtain a slightly simpler Clifford–Goldstein algorithm that yields a basis B of  F that maximizes the value .

Abstract. We prove that every verbally closed subgroup of a free group F of a finite rank is a retract of F .

Abstract. Baker and Riley gave an inclusion of a free group of rank 3 in a hyperbolic group for which the Cannon–Thurston map is not well-defined. By using their result, we show that every non-elementary hyperbolic group can be included in some hyperbolic group in such a way that the Cannon–Thurston map is not well-defined. In fact we generalize their result to every non-elementary relatively hyperbolic group.

Abstract. An element w of a Weyl group W is called elliptic if it has no eigenvalue 1 in the standard reflection representation. We determine the order of any representative  g in a semisimple algebraic group G of an elliptic element w in the corresponding Weyl group W . In particular if w has order d and G is simple of type different from C n or  F 4 , then g has order d in G .

Abstract. For , let  be the set of partitions of Ω given by the cycles of elements of  G . Under the refinement order, admits join and meet operations. We say that G is join- or meet-coherent if  is join- or meet-closed, respectively. The centralizer in  of any permutation  g is shown to be meet-coherent, and join-coherent subject to a finiteness condition. Hence if G is a centralizer in S n , then is a lattice. We prove that wreath products, acting imprimitively, inherit join-coherence from their factors. In particular automorphism groups of locally finite, spherically homogeneous trees are join-coherent. We classify primitive join-coherent groups of finite degree, and also join-coherent subgroups of S n normalizing an  n -cycle. We show that if is a chain, then there is a prime  p such that G acts regularly on each of its orbits as a subgroup of the Prüfer p -group, with G being isomorphic to an inverse limit of these subgroups.

Abstract. We characterize groups which act locally-primitively on a complete bipartite graph. The result particularly determines certain interesting factorizations of groups.

Abstract. The commuting graph of a group G is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. It was conjectured by Jafarzadeh and Iranmanesh that there is a universal upper bound on the diameter of the commuting graphs of finite groups when the commuting graph is connected. In this paper we determine upper bounds on the diameter of the commuting graph for some classes of groups to rule them out as possible counterexamples to this conjecture. We also give an example of an infinite family of groups with trivial centre and diameter 6. The previously largest known diameter for an infinite family was 5 for S n .

Abstract. Let G be a finite 2-group in which every two-generated subgroup has a cyclic commutator subgroup. We prove that the derived length of G is bounded by 4, solving problem 17.46 from the Kourovka Notebook.

Abstract. We establish necessary conditions under which is contained in a unique maximal subgroup H for a cyclic Sylow p -subgroup P of a quasisimple group G . We then use these results to establish necessary and in most cases sufficient conditions for the special case in which H is a p -local subgroup. Groups satisfying these hypotheses (including the p -local hypothesis) are precisely the groups possessing unfaithful minimal Heilbronn characters , and are relevant to the study of Artin's conjecture on the holomorphy of L -series. Moreover, since in this case is necessarily strongly p -embedded in G , this work complements existing results pertaining to strongly p -embedded subgroups of groups with noncyclic Sylow p -subgroups.