Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito-Michler Theorem asserts that if a prime p does not divide the degree of any χ ∈ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a realvalued version of this theorem, where instead of Irr(G) we only consider the subset Irr rv (G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr rv (G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained.

Let G be a permutation group on a finite set W. If G does not involve A n for n^5, then there exist two disjoint subsets of W such that no Sylow subgroup of G stabilizes both and four disjoint subsets of W whose stabilizers in G intersect trivially.Let G be a permutation group on the finite set W. We consider the following question: what is the least number of parts for a partition of W that is fixed (setwise) just by the trivial permutation in G ?Clearly, if G Sym W (resp. G Alt W) we can not take less than jWj (resp. jWj À 1) parts. But, if no symmetric or alternating group (on more than four letters) is involved in G, we will show that it is possible to find such a partition with at most 5 components (Corollary 6) and that the constant 5 is best possible (Remark 4).Our main tools are Theorem 2, which is an easy generalization of the ideas in [3, Theorem 2], and some machine computations (using GAP [2]) on a list of exceptional primitive groups from [4].Among other things, it is proved (Corollary 3) that, given a permutation group G on a finite set W, such that G does not involve A n for n^5, there exist two disjoint subsets of W that are fixed by no Sylow subgroup of G; a result that proves to be useful in finding large orbits in imprimitive module actions.We note that, through [4], our results rely on the classification of finite simple groups.Preliminaries. Let G be a permutation group on a set W (we also denote that by GY W). In the following, we always assume that W is a finite set. Our notation will mainly follow [1].Mathematics Subject Classification (1991): 20B99. 1 ) Ricerca svolta nellambito del progetto MURST ªTeoria dei gruppi e applicazioniº.

Let G be a finite group and p a prime number. We say that an element g in G is a vanishing element of G if there exists an irreducible character χ of G such that χ (g) = 0. The main result of this paper shows that, if G does not have any vanishing element of p-power order, then G has a normal Sylow p-subgroup. Also, we prove that this result is a generalization of some classical theorems in Character Theory of finite groups.

Abstract. 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.

Abstract. In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ⊆ GL(V ) acting completely reducibly on a vector space V : if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n, m > 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor.In the more general context of finite primitive permutation groups G, we show that coprime non-identity subdegrees are possible if and only if G is of O'Nan-Scott type AS, PA or TW. In a forthcoming paper we will show that, for a finite primitive permutation group, a set of pairwise coprime subdegrees has size at most 2. Finally, as an application of our results, we prove that a field has at most 2 finite extensions of pairwise coprime indices with the same normal closure.