Orbits and Stabilizers for Solvable Linear Groups

ABSTRACTWe define the position of an irreducible complex character of a finite group as an alternative to the degree. We then use this to define three classes of groups: position reducible (PR)-groups, inductively position reducible (IPR)-groups and weak IPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketa inequality (ie, that the derived length of the group is at most the number of degrees of irreducible complex characters of the group), and we show that any M-group is a weak IPR-group. We also show that even though PR-groups need not be solvable, they cannot be perfect.

Simple classical groups of Lie type are determined by their character degrees

Let G be a finite group and let be the set of degrees of complex irreducible characters of G. An important program in character theory is to obtain the structure of G when is known. In this article, we consider a dual version of this approach. Indeed, we consider non-solvable groups with five character degrees and present a description of the set of character degrees for such groups. As an application of this result, it is shown that a finite group with five character degrees and containing no prime power must be a solvable group.

Let G be a finite group. I. M. Isaacs and G. Seitz have conjectured that if G is a solvable group, then Taketa’s inequality holds, where is the set of irreducible character degrees of G and is the derived length of G. In this paper, we show that this inequality holds if G is a solvable Frobenius group. Also, we prove that if for some normal Sylow p-subgroup P of G, then . Moreover, we investigate this conjecture for some sets of the real-valued irreducible character degrees.

Let G be a finite group and let w A IrrðGÞ lie in a p-block of full defect for an odd prime p. We investigate mQðwÞp, the p-part of the rational Schur index of w. At the onset of this work, it was suspected that these Schur indices would not be divisible by p, and in Section 2 below, we show this to be the case when G is either supersolvable or p-nilpotent. Further motivation came from the situation in which G has a normal Sylow p-subgroup. Of course in this case all irreducible characters of G lie in p-blocks of full defect. Recall that a p-hyperelementary subgroup is a subgroup with a cyclic, normal p-complement and that by results of Brauer, Witt, and Berman G has a p-hyperelementary subgroup H with z A IrrðHÞ such that mQðwÞp divides mQðzÞ. If the Sylow p-subgroup of G is normal, all p-hyperelementary subgroups of G are nilpotent. Hence, the Schur indices of their irreducible characters are at most 2, and it follows that p does not divide mQðwÞ: One final case is that in which the Sylow p-subgroups of G are abelian. In this situation, Brauer’s height zero conjecture implies that the degrees of the irreducible characters in p-blocks of full defect are not divisible by p. Thus their Schur indices are not divisible by p either. However, in Section 3 we fix an a A N and construct a solvable group G with irreducible character w lying in the principal p-block of G=KerðwÞ and satisfying mQðwÞ 1⁄4 p. It is crucial to regard w as a faithful irreducible character of G=KerðwÞ, as one can always regard an irreducible character c of a group H as an irreducible character of a larger group G which surjects onto H. This does not a¤ect the Schur index of c, but it can easily be set up to ensure c lies in the principal p-block of G: The group SLð2; 3Þ has a rational-valued irreducible character w such that

Let G be a finite group and let p be a prime. In this paper, we classify all finite quasisimple groups in which the degrees of all irreducible p-Brauer characters are prime powers. As an application, for a fixed odd prime p, we classify all finite nonsolvable groups with the above-mentioned property and having no nontrivial normal p-subgroups. Furthermore, for an arbitrary prime p, a complete classification of finite groups in which the degrees of all nonlinear irreducible p-Brauer characters are primes is also obtained.

Let \(G\) be a finite group. The character degree graph of \(G\), which is denoted by \(\Gamma (G)\), is the graph whose vertices are the prime divisors of the character degrees of the group \(G\) and two vertices \(p_1\) and \(p_2\) are joined by an edge if \(p_1p_2\) divides some character degree of \(G\). In this paper we prove that the simple group \(\mathrm{PSL}(2,p^2) \) is uniquely determined by its character degree graph and its order. Let \(X_1(G)\) be the set of all irreducible complex character degrees of \(G\) counting multiplicities. As a consequence of our results we prove that if \(G\) is a finite group such that \(X_1(G)=X_1(\mathrm{PSL}(2,p^2) )\), then \(G\cong \mathrm{PSL}(2,p^2) \). This implies that \(\mathrm{PSL}(2,p^2) \) is uniquely determined by the structure of its complex group algebra.

Let G be a solvable group with five character degrees. We show that the derived length of G is at most 5. This verifies that the Taketa inequality, dl(G)≤|cd(G)|, is valid for solvable groups with at most five character degrees.

Let G be a finite group and cd(G) be the set of all irreducible complex character degrees of G without multiplicities. The aim of this paper is to propose an extension of Huppert’s conjecture from non-Abelian simple groups to almost simple groups of Lie type. Indeed, we conjecture that if H is an almost simple group of Lie type with cd(G)=cd(H), then there exists an Abelian normal subgroup A of G such that G/A≅H. It is furthermore shown that G is not necessarily the direct product of H and A. In view of Huppert’s conjecture, we also show that the converse implication does not necessarily hold for almost simple groups. Finally, in support of this conjecture, we will confirm it for projective general linear and unitary groups of dimension 3.

Finite Solvable Groups in Which Only Two Nonlinear Irreducible Characters Have Equal Degrees

Let G be a finite group. The degree (vertex) graph Γ(G) attached to G is a character degree graph. Its vertices are the degrees of the nonlinear irreducible complex characters of G, and different vertices m, n are adjacent if the greatest common divisor (m, n) > 1. In this paper, we classify all graphs with four vertices that occur as Γ(G) for nonsolvable groups G.

Let [Formula: see text] be a finite group and let [Formula: see text] be the set of all irreducible character degrees of [Formula: see text]. We consider finite groups [Formula: see text] with the property that [Formula: see text] has at most two composite members. We derive a bound 6 for the size of character degree sets of such groups. There are examples in both solvable and nonsolvable groups where this bound is met. In the case of nonsolvable groups, we are able to determine the structure of such groups with [Formula: see text].

Let A be a finite group and let n be the number of primes, including multiplicities, which divide the order of A. Suppose further that G is a finite solvable group which admits A as a fixed-point-free operator group and such that (i A I, G|I ) = 1. Under these assumptions it has been conjectured that l(G), the nilpotent length of G, is at most n, while lp(G), the p-length of G, is at most [(n+1)/2]. These inequalities have been verified and shown to be best-possible in certain special cases ( [3 ][5], [1]). It is the purpose of this note to show that these conjectured upper bounds are actually obtained provided A is solvable. Specifically we will prove

Let \(G\) be a finite solvable group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). A fundamental result by P.P. Palfy asserts that the complement $\bar{\Delta}(G)$ of the graph \(\Delta(G)\) does not contain any cycle of length \(3\). In this paper we generalize Palfy's result, showing that $\bar{\Delta}(G)$ does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of \(\Delta(G)\) can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if \(n\) is the clique number of \(\Delta(G)\), then \(\Delta(G)\) has at most \(2n\) vertices. This confirms a conjecture by Z. Akhlaghi and H.P. Tong-Viet, and provides some evidence for the famous \emph{\(\rho\)-\(\sigma\) conjecture} by B. Huppert.

Finite groups in which all the nonlinear irreducible characters have equal degrees were described by Isaacs, Passman, and others.The purpose of this article is to consider the other extreme, namely, to characterize all finite groups in which all the nonlinear irreducible characters have distinct degrees.