Non-solvable groups whose non-linear character degrees have the same number of different prime divisors

Abstract: By a result of Noritzsch, a finite solvable group whose non-linear character degrees have the same set of prime divisors is meta-abelian. In this note we investigate finite non-solvable groups whose non-linear character degrees have the same number of different prime divisors, and show that up to an abelian direct factor, such groups are exactly $L_2(4), L_2(8), A_7, S_7$, the central product of a cyclic $3$-group with $3.A_7$, or the semi-direct product of $A_7$ by a cyclic $2$-group $\langle a\rangle$ such that $a$ non-trivially acts on $A_7$ by conjugation. As consequence, we show that only the primes $2,3,5,7$ may occur as prime divisors of their irreducible character degrees, and that Huppert's $ρ$-$σ$ conjecture holds for them.

• The paper establishes a complete classification of finite non-solvable groups whose non-linear character degrees share an identical count of prime divisors.
• It employs group-theoretic methods including Clifford theory and computational GAP techniques to identify groups such as L2(4), L2(8), A7, and S7.
• The findings confirm Huppert's ρ-σ conjecture for these groups, setting sharp structural constraints on the possible prime divisors in character degrees.

Characterization of Finite Non-Solvable SNPD-Groups

Introduction

This paper addresses the structural characterization of finite non-solvable groups with a specific restriction on their complex irreducible character degrees, namely, those for which all non-linear character degrees share the same number of different prime divisors. Building upon results in the literature regarding the constraints that character degrees impose on group structure, the authors identify, up to a direct factor with an abelian group, all finite non-solvable groups with this property. The findings establish precise group catalogues and confirm Huppert's $\rho$-$\sigma$ conjecture for these groups.

Background and Definitions

A finite group $G$ is studied through the degree set ${\rm cd}(G)$ of its complex irreducible characters, and the set $\pi(n)$ of distinct prime divisors of an integer $n$. The main definition introduced is:

• SNPD-group: $G$ is an SNPD-group if, for all $1 \neq \chi(1) \in {\rm cd}(G)$, the number $|\pi(\chi(1))|$ is constant (i.e., all non-linear irreducible character degrees have the same number of different prime divisors).

Prior work is leveraged, particularly O. Manz's classification for solvable groups with prime-power character degrees and Noritzsch’s result that any finite solvable group with non-linear character degrees sharing the same set of prime divisors is meta-abelian. The paper extends these investigations into the non-solvable case.

Main Theorems and Classification

The authors deliver two main results. First, they classify the almost simple SNPD-groups:

• Theorem 1: An almost simple finite group is an SNPD-group if and only if it is isomorphic to one of $L_2(4)$, $\sigma$0, $\sigma$1, or $\sigma$2.

Second, the full structure of finite non-solvable SNPD-groups is determined:

• Theorem 2: A finite non-solvable group $\sigma$3 is an SNPD-group if and only if $\sigma$4, where $\sigma$5 is abelian and $\sigma$6 is one of:
  • $\sigma$7, $\sigma$8, $\sigma$9, or $G$0;
  • The central product of a cyclic $G$1-group with $G$2;
  • The semidirect product of $G$3 by a cyclic $G$4-group $G$5 such that $G$6.

In each case, the authors provide group-theoretic reasoning anchored to subgroup structure, Clifford theory, and character-theoretic invariants, exploiting both computational data from GAP and known results from character theory.

Proof Overview and Techniques

A rigorous exploration of the possibilities for the simple component (socle) of a non-solvable group is conducted, distinguishing the alternating groups, simple groups of Lie type, and sporadic groups (including the Tits group). Key elements include:

• Sporadic and Tits Simple Groups: Eliminated by explicit computation showing their character degrees do not satisfy the required property.
• Simple Groups of Lie Type: The presence of a Steinberg character with prime power degree allows reduction to $G$7 and $G$8 by an existing classification result [Wi87].
• Alternating Groups: Analyzed using hook formula and combinatorial properties of character degrees; only $G$9 and ${\rm cd}(G)$0 are found to satisfy the SNPD condition due to constraints on the prime divisors as ${\rm cd}(G)$1 increases.
• Almost Simple Extensions: Clifford theory and subgroup analysis are used to argue about possible extensions and direct product decompositions with abelian normal subgroups, enforcing that non-trivial extensions must be either split or factor through abelian central factors.

Technical lemmas regarding the direct factor property of maximal order elements in abelian ${\rm cd}(G)$2-groups are supplied for completeness, as they underpin steps in the final group construction.

Character Degree Implications and the ${\rm cd}(G)$3-${\rm cd}(G)$4 Conjecture

A corollary asserts:

• For any non-solvable SNPD-group, ${\rm cd}(G)$5 and ${\rm cd}(G)$6 or ${\rm cd}(G)$7.

This containment explicitly bounds the set of possible prime divisors of irreducible character degrees. It affirms that Huppert's ${\rm cd}(G)$8-${\rm cd}(G)$9 conjecture—stating that $\pi(n)$0 can be bounded in terms of $\pi(n)$1, and $\pi(n)$2 if $\pi(n)$3 is solvable—holds for the class under discussion, since $\pi(n)$4 and $\pi(n)$5 for these groups.

Implications and Future Developments

This classification closes a line of inquiry regarding the possible structure of finite non-solvable groups constrained by the uniformity of prime divisor counts across their non-linear character degrees. The strong exclusion of sporadic and most groups of Lie type is notable, reinforcing the rigidity imparted by character degree data. The explicit catalogue offers a reference point for further studies on the relation between character degrees and group structure, particularly for questions regarding extensions, maximal subgroups, and associated invariants.

Future work may investigate analogous conditions with relaxed divisor constraints, other classes of non-linear characters, or analogous results in infinite or pro-finite group settings. Connections to automorphism groups, character graphs, and block theory may also yield additional structural characterizations.

Conclusion

This paper establishes a rigorous and exhaustive classification of finite non-solvable SNPD-groups, identifying a sharply delimited collection derived from well-understood simple and almost simple groups. The results confirm that only small, concrete groups arise, and that the interplay between character degree data and group structure is sharply constraining in the non-solvable context. The findings reinforce the deep connection between representation theory and group structure, and provide a definitive answer to the specific inquiry motivating the study.