In this paper, we study groups for which if 1 < a < b are character degrees, then a does not divide b. We say that these groups have the condition no divisibility among degrees (NDAD). We conjecture that the number of character degrees of a group that satisfies NDAD is bounded and we prove this for solvable groups. More precisely, we prove that solvable groups with NDAD have at most four character degrees and have derived length at most 3. We give a group-theoretic characterization of the solvable groups satisfying NDAD with four character degrees. Since the structure of groups with at most three character degrees is known, these results describe the structure of solvable groups with NDAD.

If W is a H-module for a group H and if S is a transitive permutation group of degree n, then the wreath product K of H by S acts naturally on a module V of dimension n times that of W. Here, we give an exact count of the number of regular K-orbits on V. This can be expressed in a formula dependent only on the number of regular orbits of H on W and the permutation action of S on a set of n elements. This situation arises frequently, e.g. if V is an irreducible G-module for a finite group G and V is not quasi-primitive, then G is isomorphic as a linear group to a subgroup of a wreath product K (as above). The number of regular G-orbits is at least vertical bar K : G vertical bar times the number of regular K-orbits.

If P is a Sylow-p-subgroup of a finite p-solvable group G, we prove that G' boolean AND N-G(P) subset of or equal to P if and only if p divides the degree of every irreducible non-linear p-Brauer character of G. More generally if pi is a set of primes containing p and G is pi-separable, we give necessary and sufficient group theoretic conditions for the degree of every irreducible non-linear p-Brauer character to be divisible by some prime in pi. This can also be applied to degrees of ordinary characters.

Suppose that $P$ is a Sylow-$p$-subgroup of a solvable group $G$. If $G$ is a transitive permutation group of degree $n$, then the number of $P$-orbits is at most $2n/(p + 1)$. This is used to prove that if $G$ is a faithful irreducible linear group of degree $n$, then the dimension of the centralizer of $P$ is at most $2n/(p + 1)$. The latter result generalizes results of Isaacs and Navarro and is also used to affirmatively answer a question ofMonasur and Iranzo regarding indices of centralizers in coprime operator groups.

In this paper, we consider elements x of a finite group G with the property that χ(x)≠0 for all irreducible characters χ of G. If G is solvable and x has odd order, we show that x must lie in the Fitting subgroup F(G).

. Let G be a finite { p,q } -solvable group for different primes p and q . Let P ∈ Syl p ( G ) and Q ∈ Syl q ( G ) be such that PQ = QP . We prove that every χ ∈ Irr( G ) of p 0 -degree has q 0 -degree if and only if N G ( P ) ⊆ N G ( Q ) and C Q (cid:48) ( P ) = 1.

Let π be a set of primes and let G be a π-separable group (all groups considered are finite). Two subsets Xπ(G) and Bπ(G) of the set Irr(G) of irreducible characters of G play an important role in the character theory of π-separable groups and particularly solvable groups. If p is prime and π is the set of all other primes, then the Bπ characters of G give a natural one-to-one lift of the Brauer characters of G into Irr(G). More generally, they have been used to define Brauer characters for sets of primes.The π-special characters of G (i.e., Xπ(G)) restrict irreducibly and in a one-to-one fashion to a Hall-π-subgroup of G. If an irreducible character χ is quasi-primitive, it factors uniquely as a product of a π-special character an a π′-special character. This is a particularly useful tool in solvable groups.

All groups considered here are finite, unless otherwise specified.By Ch (G), we denote all complex characters of G; and by IRR (G), we denote the set of those $ e Ch (G) that are irreducible.(On occasion, where it involves no loss of generality to t.he specific argument, we may say A s Ch (G) allowing the pos- sibility A 0).If a group A acts on G by automorphisms and if a defined by Za(ha) z(h) is an irreducible character of G whenever ;t is.If A is cyclic, the actions of A on IRR (G) and on the conjugacy classes of G are permutation isomorphic.Counterexamples exist for noncyclic A. We write IRRA (G) to denote the A-fixed irreducible characters of G. Now assume A acts on G by automorphisms and (I G I, IAI)= 1.Should A be solvable, G. Glauberman has defined a "natural" one-to-one correspon- dence between IRRa (G)and IRR (C)[5], if C C(A).When GI is odd, I.M. Isaacs has described a "natural" correspondence between IRRa (G)and IRR (C) [6].By "natural" we mean a map uniquely determined by the action of A on G and thus independent of choices made in an algorithm.The Odd-Order Theorem implies one of these correspondences occurs.One corollary of these correspondences is that A acts isomorphically on IRR (G) and the conjugacy classes of G (see comments preceding Theorem 5.5).Both correspondences exist precisely when GI is odd and A is solvable; and we show in this paper that the two are then identical.Let N

Let G be a nite fp; qg-solvable group for dierent primes p and q .L et P 2Sylp(G )a nd Q 2Sylq(G) be such that PQ =QP .W e prove that every 2 Irr(G )o f p 0-degree has q0-degree if and only if NG(P) NG(Q) and CQ0(P )=1 .