Volume 13, Issue 6

The directed power graph of a group G is the digraph with vertex set G , having an arc from y to x whenever x is a power of y ; the undirected power graph has an edge joining x and y whenever one is a power of the other. We show that, for a finite group, the undirected power graph determines the directed power graph up to isomorphism. As a consequence, two finite groups which have isomorphic undirected power graphs have the same number of elements of each order.

Let G be a finite group and p > 2 a prime. We show that a Sylow p -subgroup of G is self-normalizing if and only if G has no non-trivial irreducible p -Brauer character of degree not divisible by p .

The aim of this note is to classify the finite groups whose irreducible characters vanish on at most three conjugacy classes in the character table.

We show that if G is a finite group with triality S and L is the corresponding Moufang loop with P 1 , P 2 ∈ Syl p ( L ) for a ‘Sylow prime’ p then there exist p -subgroups Q 1 , Q 2 ⩽ G that are conjugate in G such that Q i is invariant under S and P i is the corresponding Moufang loop for i ∈ {1, 2}. If L is simple, or if 3 does not divide | L |, then the elements of { G ( P ) ⩽ G | P ∈ Syl p ( L )} are conjugate in G and can be permuted transitively by C G ( S ), hence |Syl p ( L )| divides | C G ( S )|.

Let G be a finite group. A subgroup A of G is said to be S -quasinormal in G if AP = PA for all Sylow subgroups P of G . The symbol H sG denotes the subgroup generated by all those subgroups of H which are S -quasinormal in G . A subgroup H is said to be S -supplemented in G if G has a subgroup T such that T ∩ H ⩽ H sG and HT = G ; see [Skiba, J. Algebra 315: 192–209, 2007].

We investigate the graph associated with the p -groups of maximal class. Our main result is a periodicity theorem which extends recent results of du Sautoy and of Eick and Leedham-Green. We also obtain further details on the shape of the graph.

This article examines lower bounds for the representation growth of finitely generated (particularly profinite and pro- p ) groups. It also considers the related question of understanding the maximal multiplicities of character degrees in finite groups, and in particular simple groups.

It is proved that any infinite Abelian group of infinite exponent admits a non-discrete reflexive group topology.

We prove, under a technical assumption, that the abstract commensurator of a group that splits over a cyclic subgroup is not finitely generated. This applies in particular to free groups, surface groups, and more generally to all torsion-free word-hyperbolic groups with infinite outer automorphism group and abelianization of rank at least 2. On the other hand, we remark that the condition on the outer automorphism group cannot be removed.

We give a new construction of polyhedra with specified links and use this to investigate SQ-universality of groups with certain presentations. In particular, we answer negatively a question of J. Howie from [Forum Math. 1: 251–272, 1989] on SQ-universality of groups with so-called special presentations.