Degrees of $p$-rational characters and normality of Sylow $p$-subgroups

Abstract: Several refinements of (the normality part of) the celebrated Itô--Michler theorem were obtained during the last two decades, in which the condition of having $p'$-degree, for a fixed prime $p$, is imposed only on some subsets of complex irreducible characters of a finite group $G$. We prove further extensions of these results, where this condition is now imposed on the irreducible characters which lie above the principal character of a Sylow $p$-subgroup and are either $p$-rational, or strongly real when $p=2$.

• The paper establishes new equivalences between p-rational irreducible character degrees and the normality of Sylow p-subgroups.
• It applies advanced techniques including Deligne–Lusztig theory, Clifford theory, and explicit character table computations to validate its results.
• The study also highlights limitations through counterexamples, refining our understanding of p-divisibility in character degrees.

$p$-Rational Character Degrees and Normality of Sylow $p$-Subgroups

Introduction and Motivation

The paper "Degrees of $p$-rational characters and normality of Sylow $p$-subgroups" (2607.01706) presents several new character-theoretic criteria for determining the normality of Sylow $p$-subgroups in finite groups. Extending earlier refinements of the Itô–Michler theorem, the authors focus on the set of $p$-rational irreducible constituents above the principal character of a Sylow $p$-subgroup, and for $p=2$, the subset of strongly real such constituents. The central results establish new equivalences between the normality of $P\in \operatorname{Syl}_p(G)$ and the absence of $p$-divisibility among the degrees of these highly-restricted irreducible characters. The rigorous classification and reduction techniques outlined in the work are grounded in deep results from the theory of finite groups, representation theory, and Deligne–Lusztig theory.

Main Results

Theorem A (General $p$0)

Theorem A provides three equivalent conditions for the normality of a Sylow $p$1-subgroup $p$2 in a finite group $p$3 in character-theoretic terms focusing on $p$4-rational irreducibles over $p$5:

1. Every $p$6-rational irreducible constituent $p$7 of $p$8 has $p$9-degree.
2. For every $p$0-rational irreducible constituent $p$1 of $p$2, and for every $p$3, $p$4.
3. $p$5 is normal in $p$6.

Analogous results (Theorem C) are proved for the $p$7-rational irreducible characters with multiplicity conditions.

Theorem B ($p$8)

For $p$9, Theorem B specializes the above to the class of strongly real irreducible constituents:

1. Every strongly real irreducible constituent $p$0 of $p$1 has odd degree.
2. For every such $p$2 and every $p$3, $p$4.
3. $p$5 is normal in $p$6.

This involves significant use of the Frobenius–Schur indicator and the structure of real and strongly real characters.

Structural Reductions and Negative Results

• Strengthened forms of these theorems, imposing conditions only on $p$7-constituents with multiplicities coprime to $p$8, are shown to be false for $p$9 by explicit counterexamples.
• The analysis of small degree or exceptional simple groups is done directly, often via explicit computation or the use of known character tables.
• General proofs are reduced to verifying certain properties for almost simple groups and their covers, leveraging significant results from the theory of unipotent and semisimple character induction.

Technical Highlights

Character-theoretic and Galois-theoretic Methods

Key to the proofs is the careful tracing of $p$0-rationality through induction, restriction, and extension of characters—even when groups have complicated automorphism structures. The authors show that the property of having $p$1-degree among a select subset of irreducibles captures a strong form of control over group structure, notably the normality of Sylow $p$2-subgroups.

Reductions to Simple and Almost Simple Groups

For the reduction to simple socle cases, the paper leverages the structure of finite simple groups (including the Classification of Finite Simple Groups), the properties of character degrees, and the action of automorphisms on character sets. The major technical devices include:

• Use of Deligne–Lusztig theory for handling characters of finite groups of Lie type.
• Construction of irreducible characters with specified vanishings at non-trivial $p$3-elements and control over their fields of values.
• Explicit computations with character tables for sporadic and alternating groups.

Application of Clifford Theory and Group Cohomology

Applications of Clifford theory, particularly the correspondences for normal subgroups and for situations involving non-trivial coprime actions, are crucial. The paper interacts deeply with group cohomological results, such as in the characterization of $p$4-rationality in extensions and the effective use of Schur–Zassenhaus and the Feit–Thompson theorem when applicable.

Implications and Theoretical Impact

The criteria advanced in this paper have substantial consequences for the study of group structure via character theory, especially in modular settings and in relation to Galois actions. The refined connection between the normality of Sylow $p$5-subgroups and the arithmetic properties of character degrees opens avenues for further investigation into deeper modular analogs of ordinary character-theoretic results.

While negative results indicate that multiplicity constraints alone on $p$6-rational and strongly real characters cannot always yield such criteria, the positive theorems illuminate a robust and precise window in the landscape of character theory for interrogating group structure. These results are expected to influence both the theoretical study of representation theory in finite groups and the computational approaches to group invariants, particularly in classification problems and conjectures surrounding local–global principles.

Future Directions

Several avenues surface for future work:

• Extending similar criteria to modular (Brauer) characters or more exotic Galois-related classes of characters.
• Generalizing the methods to infinite families of finite groups with additional local or cohomological structure.
• Investigation into analogous results for blocks of defect zero and the role of $p$7-rationality in modular representation theory.
• Expanding the computational framework for verifying such results in specific families of groups, leveraging systems like GAP or Magma.

Given the fundamental nature of the group-theoretic problems addressed, this work is poised to remain a reference point for analyses at the interface of character theory and group structure.

Conclusion

The paper establishes rigorous, highly technical, and comprehensive new equivalences relating the degrees of $p$8-rational and strongly real characters to the normality of Sylow $p$9-subgroups, extending the landscape of theorems in character theory. By combining deep theoretical techniques, explicit computational insight, and negative results delimiting the scope of possible extensions, the research charts a course for new investigations into the arithmetic of character degrees and their implications for the structural analysis of finite groups (2607.01706).