Volume 15, Issue 5

Abstract. Let G be the automorphism group of a regular right-angled building X . The “standard uniform lattice” is a canonical graph product of finite groups, which acts discretely on X with quotient a chamber. We prove that the commensurator of is dense in G . This result was also obtained by Haglund (2008). For our proof, we develop carefully a technique of “unfoldings” of complexes of groups. We use unfoldings to construct a sequence of uniform lattices , each commensurable to , and then apply the theory of group actions on complexes of groups to the sequence . As further applications of unfoldings, we determine exactly when the group G is nondiscrete, and prove that G acts strongly transitively on X .

Abstract. Let G be an infinite locally compact group and let be a cardinal satisfying for the weight of G . It is shown that there is a closed subgroup N of G with . Sample consequences are: (1) Every infinite compact group contains an infinite closed metric subgroup. (2) For a locally compact group G and a cardinal satisfying , where is the local weight of G , there are either no infinite compact subgroups at all or there is a compact subgroup N of G with . (3) For an infinite abelian group G there exists a properly ascending family of locally-quasiconvex group topologies on G , say, , such that .

Abstract. We show that every quasisimple sporadic group apart from the Mathieu groups and is a strongly real Beauville group. We further show that none of the almost simple sporadic groups or any of the groups of the form or are mixed Beauville groups.

Abstract. A p -group is called an group if all of its non-normal cyclic subgroups have index no more than p m in their normalizers. In this paper we prove that the order of a non-Dedekind group cannot exceed when . We also completely classify non-Dedekind groups for .

Abstract. Let G be a finite group and let be the set of elements of primary, biprimary and triprimary order of G . We prove the following statement: if the conjugacy class sizes of are exactly with and , then G is nilpotent and for some prime q . Some known results are generalized.

Abstract. Given either a simple algebraic group or a finite group of Lie type, of rank at least 2, and a maximal parabolic subgroup, we determine which non-trivial unipotent classes have the property that their intersection with the parabolic subgroup is contained within its unipotent radical. Such classes are rare; listing them provides a basis for inductive arguments.

Abstract. Yakov Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR -subgroup (Normal Restriction) if, whenever , , where K G is the normal closure of K in G . In this article we characterize the class of finite solvable groups in which every subnormal subgroup is normal in terms of NR -subgroups. We also give similar characterizations of the classes of finite solvable groups in which every subnormal subgroup is permutable or s -permutable. Moreover we provide some sufficient conditions for the supersolvability and p -nilpotency of finite groups.