On degrees of minimal invariant characters

Abstract: It is well known that finite groups with exactly two character degrees have an abelian derived subgroup and, consequently, are solvable. Let $G$ be a finite group and $N$ a normal subgroup of $G$. In this paper, we prove that normal subgroups possessing exactly two degrees of minimal $G$-invariant characters are solvable. Furthermore, it is shown that if these degrees are ${1, f}$ for some integer $f$, then either $f$ is a prime power or the commutator subgroup $[N,G]$ is abelian. Whether $[N, G]$ is abelian when $f$ is a prime power remains an open problem. Specifically, we prove that this holds when $f=p$.

• The paper proves that if Mcd_G N = {1, f}, then the normal subgroup N is solvable.
• It employs Clifford theory and induced characters to relate orbit sizes and character degrees to subgroup structure.
• The results establish that either f is a prime power or [N, G] is abelian, generalizing classical Burnside and Thompson theorems.

Minimal G-Invariant Character Degrees and Solvability of Normal Subgroups

Introduction and Preliminaries

The paper "On degrees of minimal invariant characters" (2606.14333) investigates the structure of finite groups through the lens of character theory, focusing on minimal $G$-invariant characters of normal subgroups. Given a finite group $G$ and $N \unlhd G$, the action of $G$ by conjugation on $\mathrm{Irr}(N)$ leads to the construction of minimal $G$-invariant characters $\widehat{\theta}$, defined as the sum over the $G$-orbit of $\theta \in \mathrm{Irr}(N)$. The degree set of these, $$\mathrm{Mcd}_G N = \{ \widehat{\theta}(1) \mid \theta \in \mathrm{Irr}(N) \}$$, plays a central role.

Previously, groups with two character degrees were known to have abelian derived subgroups and to be solvable, a feature leveraged in classical results by Burnside and Thompson. The authors advance these classical themes by establishing structural consequences for normal subgroups $G$0 with restricted minimal $G$1-invariant character degrees, focusing on the case where $G$2.

Main Structural Results

The principal contribution is the proof that if a normal subgroup $G$3 of $G$4 satisfies $G$5, then $G$6 is solvable (Theorem A). This result broadens the classical character degree techniques, as the analogous property for $G$7 with two irreducible character degrees is well established, but the situation for normal subgroups is more nuanced since $G$8 may admit more than two irreducible character degrees, while having only two minimal $G$9-invariant ones.

A key observation underpinning these results is the extension of the Burnside and Thompson theorems to the context of minimal $N \unlhd G$0-invariant character degrees. The authors carefully analyze the interplay between the sizes of $N \unlhd G$1-orbits on $N \unlhd G$2 and the behavior of $N \unlhd G$3’s commutator and structure.

A conjecture is proposed and partially resolved: if $N \unlhd G$4, then $N \unlhd G$5 is abelian (Conjecture B). The only open case is where $N \unlhd G$6 is a prime power; the paper proves the conjecture when $N \unlhd G$7 for a prime $N \unlhd G$8 (Theorem D).

The full structural implications are codified in Theorem C:

• If $N \unlhd G$9, then either $G$0 is a prime power or $G$1 is abelian.
• If $G$2 is a $G$3-group, then $G$4 is hypercentral in $G$5 and $G$6 (the Frattini subgroup).
• If $G$7 is not a prime power, then $G$8 must be abelian.

These theorems utilize a careful reduction process based on the structure of minimal normal subgroups and deploy results from classical character theory, including Clifford theory and detailed knowledge of character extensions and character degrees for nonabelian simple groups.

Fine Structure and Group-Theoretic Consequences

A considerable component of the paper is devoted to analyzing the possible configurations of $G$9 under the two-degree minimal $\mathrm{Irr}(N)$0-invariant hypothesis. Proposition 12.3 refines the understanding of the quotient $\mathrm{Irr}(N)$1 and the relation between the character degrees and the action of $\mathrm{Irr}(N)$2:

• In the abelian case, $\mathrm{Irr}(N)$3 is an abelian $\mathrm{Irr}(N)$4-group with $\mathrm{Irr}(N)$5 elementary abelian.
• In the nonabelian case, either $\mathrm{Irr}(N)$6 is a $\mathrm{Irr}(N)$7-group with $\mathrm{Irr}(N)$8 elementary abelian, where certain explicit relations among $\mathrm{Irr}(N)$9, $G$0, and group invariants hold; or $G$1 is a Frobenius group with abelian complement and elementary abelian kernel.

The use of such fine structure allows the authors to rule out non-solvable configurations systematically and identify the (mostly restrictive) ways in which $G$2 can support only two minimal $G$3-invariant character degrees.

Importantly, the paper highlights that the existence of only two degrees for the minimal $G$4-invariant characters does not force $G$5 to be nilpotent (see explicit GAP examples provided), distinguishing the result from other character-theoretic theorems about classes and degrees.

Numerical and Exceptional Results

Prominent numerical results include:

• If $G$6 for a prime $G$7, then $G$8 is abelian.
• If $G$9 is a $\widehat{\theta}$0-group and all minimal $\widehat{\theta}$1-invariant character degrees are powers of $\widehat{\theta}$2, then $\widehat{\theta}$3 must be hypercentral in $\widehat{\theta}$4, with $\widehat{\theta}$5.
• If $\widehat{\theta}$6 is not a prime power, $\widehat{\theta}$7 is necessarily abelian.

Numerous explicit group-theoretic examples (with GAP identifiers) are provided to demonstrate the necessity of various hypotheses, and exceptions to nilpotency or hypercentrality in the absence of suitable constraints.

Methodology and Character-Theoretic Techniques

The proofs rely heavily on Clifford theory, the properties of inertia subgroups, induced characters, and the structure of $\widehat{\theta}$8-orbits on $\widehat{\theta}$9. Ramification numbers, Schur indices, and detailed arguments using the orthogonality relations and classical theorems (such as Itô-Michler and Taketa) are systematically leveraged.

Burnside’s and Thompson’s results are generalized in the context of minimal $G$0-invariant character degrees, allowing for a dual perspective between conjugacy class actions and character-theoretic invariants. The authors delineate the limitations of naively applying the ordinary Burnside theorem in this context.

Further, an analysis of the lower central $G$1-series for $G$2 and its hypercentrality is given, showing that when $G$3 inherits additional group-theoretic properties (as in the $G$4-group case), this series terminates rapidly.

Implications and Future Directions

The work establishes that group structure can be effectively analyzed—up to solvability and near-nilpotency—based on minimal $G$5-invariant character degree data, extending the reach of character degree methods well beyond the ambient group $G$6 to its normal subgroups. The techniques developed open further directions:

• Analyzing higher numbers of minimal $G$7-invariant degrees, and associated influence on nilpotency or other structural invariants.
• Investigating the open case in Conjecture B for $G$8 with $G$9.
• Extending the methods to other types of group actions or to modular character theory.

The discussion and numerous explicit constructions provide guidance for further explorations with computational algebra systems such as GAP, and for the integration of character-theoretic and group-theoretic invariants in the classification and analysis of finite groups.

Conclusion

This paper demonstrates that the set of minimal $\theta \in \mathrm{Irr}(N)$0-invariant character degrees for a normal subgroup $\theta \in \mathrm{Irr}(N)$1 of a finite group $\theta \in \mathrm{Irr}(N)$2 imposes strong constraints on the structure of $\theta \in \mathrm{Irr}(N)$3. When $\theta \in \mathrm{Irr}(N)$4, the subgroup $\theta \in \mathrm{Irr}(N)$5 is necessarily solvable, and, under further numerical constraints on the nontrivial degree, $\theta \in \mathrm{Irr}(N)$6 is abelian or $\theta \in \mathrm{Irr}(N)$7 is hypercentral in $\theta \in \mathrm{Irr}(N)$8. These results enrich the interface between character theory and group structure, generalizing and refining classical theorems, and laying the groundwork for further character-theoretic investigations of subgroup structure in finite groups.