Volume 18, Issue 2

Let 𝒲 be a set of nilpotent Hall subgroups of a finite group G . We say that 𝒲 is a Wielandt set if for each prime p dividing | G | there is exactly one subgroup W ∈ 𝒲 such that p divides | W |. In this paper, we study the influence of Wielandt sets on the structure of G . In particular, we give conditions under which a normal subgroup of a finite group is hypercyclically embedded.

In this paper, we present results for a group theoretic analog to a parameter defined in terms of irreducible character degrees. Let G be a finite group and k be the size of the largest conjugacy class of G . The parameter e is defined by the equation |G|=k(k+e)${|G|=\sqrt{k}(\sqrt{k}+e)}$. We show that |G|≤(p/(p-1)2)e2${|G| \le (p/(p-1)^2)e^2}$ where p is the smallest prime dividing the order of G and classify groups that satisfy this bound.

The product replacement graph Γ k ( G ) is the graph on the generating k -tuples of a group G , with edges corresponding to Nielsen moves. We prove the exponential growth of product replacement graphs Γ k (𝔾 ω ) of Grigorchuk groups, for k ≥ 5.

We introduce the notion of the k -closure of a group of automorphisms of a locally finite tree, and give several examples of the construction. We show that the k -closure satisfies a new property of automorphism groups of trees that generalises Tits' Property P . We prove that, apart from some degenerate cases, any non-discrete group acting on a tree with this property contains an abstractly simple subgroup.

A group G is called logically cyclic if it has an element s such that every element of G can be defined by a first-order formula with parameter s . The aim of this paper is to investigate the structure of such groups.

We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condition for the Gromov boundaries of any two hyperbolic groups to be homeomorphic (in terms of the topology of the boundaries of factors in terminal splittings over finite subgroups).

Let G be a group and let κ be a cardinal. A subset A of G is called left (right) κ-large if there exists a subset F of G such that | F | < κ and G = F A ( G = A F ). We say that A is κ-large if A is left and right κ-large. It is known that every infinite group G can be partitioned into countably many ℵ 0 -large subsets. On the other hand, every amenable (in particular, Abelian) group G cannot be partitioned into more than ℵ 0 ℵ 0 -large subsets. We prove that every group G of cardinality κ can be partitioned into κ ℵ 1 -large subsets and every free group F κ in the infinite alphabet κ can be partitioned into κ 4-large subsets but cannot be partitioned into three 3-large subsets.

We study the situation of a finite group G having a subgroup H with the property that each of its nonprincipal irreducible (complex) characters induce to G as a sum of irreducible characters all having the same degree. When this happens, the subgroup H either contains, or is contained in, the commutator subgroup G ' of G . In any case, the normal closure H G is always proper in G whenever H is proper (even when G is perfect), and H G is solvable whenever this normal closure is a proper subgroup of G '.

In this article we study the vertices of simple modules for the symmetric groups in prime characteristic p . In particular, we complete the classification of the vertices of simple F 𝔖 n -modules labelled by hook partitions.