Galois Version of the Itô–Michler Theorem

• The theorem refines classical character criteria by replacing the universal condition with a Galois-invariant requirement on irreducible characters.
• It leverages advanced block theory, cohomological methods, and Clifford theory to deduce structural properties of Sylow p-subgroups.
• Applications include establishing abelianness and direct product decompositions in finite groups through precise Galois action analysis.

The Galois version of the Itô–Michler theorem constitutes a major refinement of two central results in local–global character theory: the classical Itô–Michler theorem and Brauer's Height Zero Conjecture. It replaces a universal character-degree condition by a Galois-invariant one, thereby drastically weakening the requirements for strong structural conclusions about Sylow $p$-subgroups. Specifically, it identifies finite group structural properties from data involving only those irreducible characters fixed by a precise group of Galois automorphisms. The formulation and proof leverage deep block-theoretic, cohomological, and character-theoretic methods, and have substantial implications for the landscape of local–global conjectures in finite group theory (Malle et al., 2022, Moretó et al., 23 Nov 2025).

1. Classical Framework and Galois Extensions

The classical Itô–Michler theorem asserts: for a finite group $G$ and a prime $p$, the following are equivalent—

1. Every irreducible complex character of $G$ has degree prime to $p$;
2. The Sylow $p$-subgroup $P$ is normal and abelian in $G$.

This theorem provides a bridge between global character degree information and local Sylow subgroup structure. It is used, for instance, to deduce direct product decompositions $G=O_{p'}(G)\times P$ by examining divisibility of character degrees (Malle et al., 2022).

The Galois extension of the Itô–Michler theorem replaces the global requirement on all irreducible characters with a restriction solely on those characters invariant under a specified subgroup of Galois automorphisms. This generalization intersects fundamentally with block theory and Galois actions, deeply involving the theory of field automorphisms on character values (Malle et al., 2022, Moretó et al., 23 Nov 2025).

2. Technical Background: Blocks, Heights, and Galois Actions

For a finite group $G$ and prime $p$, the set $\Irr(G)$ of irreducible complex characters decomposes according to the block structure of the group algebra. The principal $p$-block $B_0(G)$ is the unique block whose defect group is a Sylow $p$-subgroup of $G$. A character $\chi\in\Irr(B_0(G))$ has height zero exactly when $\chi(1)$ is not divisible by $p$ ($\chi(1)_p=1$).

Galois automorphisms act on character values: if $\sigma\in\Gal(\Qab/\Q)$, then

$$\chi^\sigma(g) = \sigma(\chi(g)),\quad\forall g\in G.$$

For a subgroup $\mathcal{J} \subseteq \Gal(\Qab/\Q)$ of automorphisms of order $p$ fixing all $p$-power roots of unity, the set of $\mathcal{J}$-invariant irreducible characters is defined by

$\Irr_{\mathcal{J}}(G) = \{\chi \in \Irr(G) \mid \chi^\sigma = \chi,~\forall \sigma\in\mathcal{J}\}.$

A key special case for $p=2$ employs the automorphism $o$ that fixes all $2$-power roots of unity and conjugates odd-order roots of unity. The set of $o$-invariant irreducible characters is denoted $\Irr^o(G)$ (Malle et al., 2022, Moretó et al., 23 Nov 2025).

3. Statement and Structural Description of the Galois Itô–Michler Theorem

The main structural result (Theorem B of (Moretó et al., 23 Nov 2025)) is:

Galois Itô–Michler Theorem (Moretó–Rizo–Souza):

Let $G$ be finite and $p$ any prime. If

$\Irr_{\mathcal{J}}(G) \subseteq \Irr_{p'}(G)$

(i.e., every $\mathcal{J}$-invariant irreducible character of $G$ has degree prime to $p$), then

$O^{p'}(G) = O_p(G) \times K,$

where $K\unlhd G$ satisfies:

• $O_{p'}(K)$ is solvable;
• $K/O_{p'}(K) \cong S_1 \times \cdots \times S_r$, a direct product of non-abelian simple groups of order divisible by $p$, each with no $\mathcal{J}$-invariant character of $p$-power degree in its principal $p$-block (Moretó et al., 23 Nov 2025).

For $p=2$, with $o$ as above, the theorem specializes: if every $o$-invariant irreducible character has odd degree, then the Sylow $2$-subgroup is normal and abelian (Malle et al., 2022).

4. Brauer’s Height Zero Conjecture and Galois-Driven Strengthening

Brauer’s Height Zero Conjecture (in its principal block form) states that all irreducible characters in the principal $p$-block have degree prime to $p$ if and only if the Sylow $p$-subgroups are abelian.

The Galois version, established as Theorem A in (Moretó et al., 23 Nov 2025), asserts: if every $\mathcal{J}$-invariant character $\chi$ in $B_0(G)$ has $\chi(1)_p=1$, then the Sylow $p$-subgroups are abelian. This reduces the verification from all irreducibles to a restricted class of Galois-invariant ones.

A direct consequence is that it suffices to check the coprime degree condition only on the set of rational (fixed by all automorphisms) characters to guarantee abelianness of the Sylow $p$-subgroup.

5. Proof Strategy and Critical Technical Tools

The proof leverages several mechanisms:

• Reduction to a finite group of automorphisms $\Omega$: For a finite $G$, the infinite group $\mathcal{J}$ acts via a finite quotient $\Omega\subseteq\mathcal{J}$ of order $p$, so invariance under $\mathcal{J}$ reduces to invariance under $\Omega$.
• Minimal counterexample and Clifford theory: Minimal counterexample induction reduces the group structure to $G=O^{p'}(G)$, $O_{p'}(G)=1$, with unique principal $p$-block.
• Analysis by minimal normal subgroup $N$: The cases $N$ non-abelian simple, $N$ cyclic of order $p$, or $N$ elementary abelian $p$-group are handled via Clifford theory and known classifications of $p$-exceptional groups.
• Block-theoretic correspondences: The proof uses the Third Main Theorem of Brauer for relations between principal blocks in normal subgroups and quotients, and the Clifford–Alperin–Dade correspondence to lift properties through extensions.
• CFSG-dependent theorems: In cases involving elementary abelian $p$-groups, the proof relies on the classification of primitive $p$-exceptional linear and permutation groups by Giudici–Liebeck–Praeger–Saxl–Tiep.

Technical consequences are drawn using the structure of principal blocks, the lifting of Galois-invariant characters to $G$, and the construction of non-linear $\Omega$-invariant characters in each minimal normal subgroup case (Malle et al., 2022, Moretó et al., 23 Nov 2025).

6. Corollaries, Examples, and Applications

Key consequences and illustrative cases include:

• For $G$ symmetric group $\Sym_n$ and odd prime $p$ dividing $n$, $\Irr_{\mathcal{J}}(B_0(G))$ contains the natural permutation character of degree $n-1$, divisible by $p$. The Galois–Itô–Michler hypothesis fails exactly when the Sylow $p$-subgroup is non-abelian.
• For $p$-solvable $G$ with $\Irr_{\mathcal{J}}(G)\subseteq\Irr_{p'}(G)$, $G=O_{p'}(G)\times O_p(G)$.
• For odd $p$, a parallel characterization of $p$-closed groups is given via $p$-rational irreducible characters with Brauer lifts (Malle et al., 2022).

These results translate global–local character degree conditions under Galois invariance into precise statements about group structure, refining and generalizing prior formulations.

7. Impact and Prospects for Generalization

The Galois version of the Itô–Michler theorem demonstrates that requiring only the Galois-invariant characters to have $p'$-degree is sufficient to recover, up to explicit simple group obstruction, the conclusions of the original theorem. This defines a new, sharply reduced locus of character data from which local group-theoretic information may be extracted.

Future directions include:

• Extending results from the principal block to arbitrary $p$-blocks.
• Considering broader classes of Galois automorphism groups (e.g., of order dividing specified primes).
• Applying Galois Itô–Michler techniques in the context of other local–global conjectures, such as the Alperin–McKay conjecture.

The synthesis of Galois-theoretic, block-theoretic, and character-theoretic arguments in the Galois Itô–Michler theorem creates new pathways for the paper and resolution of longstanding problems in finite group theory (Moretó et al., 23 Nov 2025, Malle et al., 2022).

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Key Papers

Title	Authors	arXiv ID
Height Zero Conjecture with Galois Automorphisms	Malle, Navarro	(Malle et al., 2022)
Height zero characters and Galois automorphisms	Moretó, Rizo, Souza	(Moretó et al., 23 Nov 2025)