Carlo Casolo

Torsion-free groups in which every subgroup is subnormal

We prove that a torsion-free group in which every subgroup is subnormal is nilpotent.

Prime divisors of irreducible character degrees and of conjugacy class sizes in finite groups

Abstract We establish linear bounds for the ρ-σ conjecture for irreducible character degrees and conjugacy class sizes of any finite group.

Nilpotent Subgroups of Groups with All Subgroups Subnormal

It is proved that in groups with all subgroups subnormal every nilpotent subgroup is contained in a normal nilpotent subgroup.

Finite groups with small conjugacy classes

We study finite groups G in which the number of distinct prime divisors of the length of the conjugacy classes is at most three. In particular we prove, under this condition, a conjecture of B. Huppert on the number of prime divisors of ÷G/Z(G)÷.

On the Fitting height of factorised soluble groups

Abstract In this paper we are concerned with finite soluble groups G admitting a factorisation G=AB

Some linear actions of finite groups with q′-orbits

{G=AB}

SOLUBLE GROUPS WITH FINITE WIELANDT LENGTH

, with A and B proper subgroups having coprime order. We are interested in bounding the Fitting height of G in terms of some group-invariants of A and B, including the Fitting heights and the derived lengths.

CONJUGACY CLASS SIZES OF CERTAIN DIRECT PRODUCTS

Abstract Let G be a finite group acting faithfully on a finite vector space M in such a way that the centralizer of every element of M contains a Sylow q-subgroup of G as a central subgroup (for a fixed prime divisor q of |G| with (q, |M|) = 1). Then G is isomorphic to a subgroup of the semi-linear group on M.

Subnormality in the join of two subgroups

It may well happen that the Wielandt subgroup of a group G is trivial; for instance,this is the case if G is the infinite dihedral group. On the other hand H. Wielandt showedin [8] that in a finite group G the socle (that is the subgroup generated by all minimalnormal subgroups) is contained in co(G). Thus any finite group has finite Wielandt length.The relation between the Wielandt length and the derived and Fitting length in afinite soluble group was first investigated by A. Camina in [2]. Recently R. Bryce and J.Cossey [1] improved on Caminas results by obtaining best possible bounds for both thederived and the Fitting length of a finite soluble group in terms of its Wielandt length.The aim of this paper is to extend these results to infinite groups. To do this we haveto restrict ourselves to the class of groups with finite Wielandt length. Extending thenotation of Bryce and Cossey [1], we denot

On the subnormalizer of a p-subgroup

We consider finite groups in which, for all primes p, the p-part of the length of any conjugacy class is trivial or fixed; obtaining a full description in the case in which for each prime divisor p of the order of the group there exists a non-central conjugacy class of p-power size. DOI: 10.1017/S0004972711002735