Silvio Dolfi

Large orbits in coprime actions of solvable groups

Let G be a solvable group of automorphisms of a finite group K. If |G| and |K| are coprime, then there exists an orbit of G on K of size at least √|G|. It is also proved that in a π-solvable group, the largest normal π-subgroup is the intersection of at most three Hall π-subgroups.

Finite groups whose noncentral commuting elements have centralizers of equal size.

(y). An important subclass of F-groups is the class of I-groups, in which all centralizers of noncentralelements are of the same order. Ito proved in [I] that I-groups arenilpotent and they are direct products of an abelian group and a groupof prime-power order. Only 49 years later, K. Ishikawa showed in [Ish]that groups in I are of class at most 3. For a simpler proof, see thepapers of A. Mann [M1] and of M. Isaacs [Is]. The F-groups wereinvestigated by J. Rebmann in [R]. He determined their strucure, upto that of F-groups which are central extensions of groups of prime-power order.Another important subclass of F-groups is the class of CA-groups,consisting of groups in which all centralizers of noncentral elementsare abelian. The CA-groups (or rather the equivalent class of M-groups) were investigated by R. Schmidt in [S] (see also [S1], Theorem9.3.12). He determined their structure up to that of CA-groups whichare central extensions of groups of prime-power order. It is very similarto the structure of F-groups.

Finite groups in whichp′-classes haveq′-length

IfG is a finite group in which every element ofp′-order centralizes aq-Sylow subgroup ofG, wherep andq are distinct primes, it is shown thatOq′ (G) is solvable,lq(G)≤1 andlp(Oq′ (G)) ≤2. Further, the structure ofG is determined to some extent.

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.

A new solvability criterion for finite groups

In 1968, John Thompson proved that a finite group G is solvable if and only if every 2-generator subgroup of G is solvable. In this paper, we prove that solvability of a finite group G is guaranteed by a seemingly weaker condition: G is solvable if, for all conjugacy classes C and D of G consisting of elements of prime power order, there exist x in C and y in D with x and y generating a solvable group. We also prove the following property of finite nonabelian simple groups, which is the key tool for our proof of the solvability criterion: if G is a finite nonabelian simple group, then there exist two prime divisors a and b of |G| such that, for all elements x, y in G with |x|=a and |y|=b, the subgroup generated by x and y is not solvable. Further, using a recent result of Guralnick and Malle, we obtain a similar membership criterion for any family of finite groups closed under forming subgroups, quotients and extensions.

THE GROUPS WITH EXACTLY ONE CLASS OF SIZE A MULTIPLE OF P

Abstract Let p be a prime. The goal of this paper is to classify the finite groups with exactly one conjugacy class of size a multiple of p.

On the size of the nilpotent residual in finite groups

Abstract Let G be a finite non-abelian group satisfying Φ(G) = 1 and denote by U the nilpotent residual of G. In this paper, we prove that if G is of odd order then , and if G is of even order not divisible by a Mersenne or a Fermat prime then . These results are best possible and the assumption Φ(G) = 1 cannot be omitted.

NONVANISHING ELEMENTS FOR BRAUER CHARACTERS

Let G be a finite group, and p a prime. We say that a p-regular element g of G is p-nonvanishing if no irreducible p-Brauer character of G takes the value 0 on g. The main result of this paper shows that if G is solvable and g ∈ G is a p-regular element which is p-nonvanishing, then g lies in a normal subgroup of G whose p-length and p′-length are both at most 2 (with possible exceptions for p ≤ 7), the bound being best possible. This result is obtained through the analysis of one particular orbit condition in linear actions of solvable groups on finite vector spaces, and it generalizes (for p > 7) some results in [3]. Introduction and preliminaries An element g of a finite group G is called a nonvanishing element if χ(g) 6= 0 for every irreducible complex character χ ∈ Irr(G); in other words, g is nonvanishing if the column corresponding to g in the character table of G contains no zero entries. In [7] M. Isaacs, G. Navarro and T. Wolf prove that if G is a finite solvable group and g ∈ G is a nonvanishing element of odd order, then g lies in the Fitting subgroup F(G) of G. Moreover, in [4] it is shown that if G is any finite group and g ∈ G is a nonvanishing element of order coprime to 6, then again g ∈ F(G). In this spirit, given a prime number p, we consider elements corresponding to columns with no zero entries in the table of Brauer characters in characteristic p. We say that a p-regular element g ∈ G is a p-nonvanishing element if φ(g) 6= 0 for every irreducible p-Brauer character φ ∈ IBrp(G). In [3] it is proved that if, for a prime p > 3, all p-regular elements of a finite group G are p-nonvanishing (a condition that implies the solvability of G), then G has p′-length at most 2. Assuming (as we may, since every p-Brauer character of G contains Op(G) in its kernel) that Op(G) = 1, this implies that every p-regular element of G lies in Op′pp′(G). The main theorem of the present paper extends this result, in the case p > 7. Theorem A. Let p be a prime number greater than 3, let G be a finite solvable group with Op(G) = 1, and let g be a p-regular element of G that is p-nonvanishing. Then g lies in Op′pp′(G), unless p ∈ {5, 7} and the order of g is divisible by 2 or 3. Example 4.1 shows that the above statement is “optimal” in some sense. Our approach to Theorem A consists in studying a related problem about linear actions

Finite Groups with Only One NonLinear Irreducible Representation

Let 𝕂 be an algebraically closed field. We classify the finite groups having exactly one irreducible 𝕂-representation of degree bigger than one. The case where the characteristic of 𝕂 is zero, was done by G. Seitz in 1968.

A NOTE ON CHARACTER RESTRICTIONS

ABSTRACT Let be an irreducible character of a finite group . If , then the degrees of the irreducible constituents of the restriction can be controlled by some conditions on and on the normalizer of in .