Alexander Moretó

Non-divisibility among character degrees

Abstract 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.

Groups with two extreme character degrees and their normal subgroups

We study the finite groups G for which the set cd(G) of irreducible complex character degrees consists of the two most extreme possible values, that is, 1 and G : Z(G)[1/2. We are easily reduced to finite p-groups, for which we derive the following group theoretical characterization: they are the p-groups such that IG : Z(G)l is a square and whose only normal subgroups are those containing G or contained in Z(G). By analogy, we also deal with pgroups such that G : Z(G)I = p2n+1 is not a square, and we prove that cd(G) = {1,pn} if and only if a similar property holds: for any N < G, either G < N or NZ(G) : Z(G) < p. The proof of these results requires a detailed analysis of the structure of the p-groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than 2, then the index of the centre is small, and in some cases we may even bound the order of G.

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 number of conjugacy class sizes and character degrees in finite -groups

In this note we prove that for any two integers r, s > 1 there exist finite p-groups G of class 2 such that | cd(G)| = r and | cs(G)| = s.

Character degrees and nilpotence class of finite -groups: An approach via pro- groups

Let S be a finite set of powers of p containing 1. It is known that for some choices of S, if P is a finite p-group whose set of character degrees is S, then the nilpotence class of P is bounded by some integer that depends on S, while for some other choices of S such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set S is class bounding if and only if p ¬∈ S. In this article we provide a new approach to this problem. Our main result shows the relevance of certain p-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets S such that p ¬∈ S.

Groups St Andrews 2001 in Oxford: Characters of p -groups and Sylow p -subgroups

The aim of this note is to present some problems and also partial results in some cases, mainly on characters of p-groups. (In the last section we deal with a problem that consists in obtaining information about characters of a Sylow p-subgroup of an arbitrary group from information about the characters of the whole group.) This survey is far from being exhaustive. The topics included are strongly influenced by the author’s interests in the last few years. There seems to be an increasing interest in the character theory of p-groups and we hope that this expository paper will encourage more research in the area. In the sixties I. M. Isaacs and D. S. Passman [17, 18] wrote two important papers that initiated the study of the degrees of the irreducible complex characters of finite groups (henceforth referred to as character degrees). The study of the influence of the set of character degrees on the structure of a group was taken up again in the eighties, in large part due to B. Huppert and his school. In particular, this has led to several papers dealing with the character degrees of important families of p-groups since the nineties (see [6, 8, 12, 28, 30, 32, 33, 34, 35, 36, 37]). Here we are mostly concerned with character degrees, but instead of studying particular families of p-groups, we intend to obtain general structural properties of groups according to their character degrees. Other problems on characters of p-groups appear in [25]. The notation is standard. All the groups considered are finite. We write cd(G) to denote the set of character degrees of a group G, b(G) the maximum of the character degrees of G, cs(G) the set of conjugacy class sizes, and c(G) and dl(G) the nilpotence class and derived length of G, respectively. The terms of the ascending Fitting series of a group G will be denoted Fi(G) and the Fitting subgroup F (G). If P is a p-group we write Ωi(P ) to denote the subgroup of P generated by the elements of order ≤ pi.

An elementary method for the calculation of the set of character degrees of somep-groups

Letq be a power of a prime numberp. In this work, we present an elementary method to find out the set of irreducible character degrees of the Sylowp-subgroups ofSp(2n,q) whenq is odd. This simplifies the proof of Previtali [4], where this result was first obtained. This method also works for the groups of unitriangular matrices over finite fields.

ZEROS OF CHARACTERS ON PRIME ORDER ELEMENTS

Suppose that G is a finite group, let χ be a faithful irreducible character of degree a power of p and let P be a Sylow p-subgroup of G. If χ(x) ≠ 0 for all elements of G of order p, then P is cyclic or generalized quaternion. * The research of the first author is supported by a grant of the Basque Government and by the University of the Basque Country UPV 127.310-EB160/98. † The second author is supported by DGICYT.Suppose that G is a finite group, let χ be a faithful irreducible character of degree a power of p and let P be a Sylow p-subgroup of G. If χ(x) ≠ 0 for all elements of G of order p, then P is cyclic or generalized quaternion. * The research of the first author is supported by a grant of the Basque Government and by the University of the Basque Country UPV 127.310-EB160/98. † The second author is supported by DGICYT.

BOUNDING THE NUMBER OF IRREDUCIBLE CHARACTER DEGREES OF A FINITE GROUP IN TERMS OF THE LARGEST DEGREE

We conjecture that the number of irreducible character degrees of a finite group is bounded in terms of the number of prime factors (counting multiplicities) of the largest character degree. We prove that this conjecture holds when the largest character degree is prime and when the character degree graph is disconnected.

Character degree graphs, blocks and normal subgroups

There are several graphs associated with the set of character degrees of a finite group that have been studied. Results on these graphs are often useful when proving results giving structural information of the group from properties of character degrees. The most commonly studied graph is the graph GðGÞ whose vertices are the prime divisors of character degrees of the group G, with two vertices joined by an edge if the product of the primes divides some character degree. This graph was introduced in [7]. However, it is often interesting in character theory to study only certain subsets of the set of character degrees of a group. For instance, the sets of degrees of the members of IrrðGjNÞ 1⁄4 fw A IrrðGÞ jNGKer wg or IrrpðGÞ 1⁄4 fw A IrrðGÞ j wð1Þ is a p-numberg have been widely studied. The graphs associated to these sets of degrees have also been studied. See [2], [3], [8]. The goal of this paper is to introduce two new graphs associated to certain subsets of character degrees and to prove that they share some of the properties of the previously studied graphs. A situation often of interest in character theory is the following. We have a normal subgroup N of a finite group G; y A IrrðNÞ and we want to study the characters of G lying over y. As usual, we write IrrðGjyÞ to denote this set of characters. Our first graph considers these characters. We define the graph GðGjyÞ as follows. Its vertices are the prime divisors of the numbers wð1Þ=yð1Þ, where w A IrrðGjyÞ. We join two vertices by an edge if the product of the two di¤erent primes divides some member of fwð1Þ=yð1Þ j w A IrrðGjyÞg. Our main result shows that the number of connected components of this graph behaves as in the previously studied situations.