Volume 15, Issue 2

Abstract. We prove that the subgroup lattices of finite alternating and symmetric groups do not contain so-called lower signalizer lattices in the class . This result is one step in a program to show that the lattices in the class are not isomorphic to an interval in the subgroup lattice of any finite group.

Abstract. We study those group rings in which the group of units is hyperbolic.

Abstract. A group is said to be a T-group if all its subnormal subgroups are normal. Let be a finite solvable T-group. It is shown that the normalizer property holds for . As a direct consequence of our result, we obtain that the normalizer property holds for finite groups all of whose Sylow subgroups are cyclic.

Abstract. In this paper, we classify the -groups all of whose proper subgroups have cyclic Frattini subgroups, for an odd prime .

Abstract. We construct, for and , a -generated -group which, in an asymptotic sense, behaves almost like a -generated free pro--group. We show that a subgroup of index needs generators, and that the subgroup growth of satisfies , where is the -generated free pro--group. To do this we introduce a new invariant for finitely-generated groups and study some of its basic properties.

Abstract. Suppose that there exists a surjective isometry between open subgroups of the groups of invertible elements in the measure algebras on LCA groups. We show that the LCA groups are topologically isomorphic to each other.

Abstract. If is a group of permutations of a set , then the suborbits of  are the orbits of point-stabilizers acting on . The cardinalities of these suborbits are the subdegrees of . Every infinite primitive permutation group with finite subdegrees acts faithfully as a group of automorphisms of a locally-finite connected vertex-primitive directed graph with vertex set , and there is consequently a natural action of  on the ends of . We show that if is closed in the permutation topology of pointwise convergence, then the structure of  is determined by the length of any orbit of  acting on the ends of . Examining the ends of a Cayley graph of a finitely-generated group to determine the structure of the group is often fruitful. B. Krön and R. G. Möller have recently generalised the Cayley graph to what they call a rough Cayley graph , and they call the ends of this graph the rough ends of the group. It transpires that the ends of are the rough ends of , and so our result is equivalent to saying that the structure of a closed primitive group whose subdegrees are all finite is determined by the length of any orbit of  on its rough ends.

Abstract. We give a new proof of a theorem of Psaltis which says that the full automorphism group of every countable rooted tree has property (FA).

Abstract. We prove that Higman's group does not embed into a metric ultraproduct of finite groups with a commutator-contractive invariant length function.