On The Eaton-Moretó Conjecture for Principal Blocks of Finite Groups

Abstract: Let $G$ be a finite group and let $p$ be a prime. If $P$ is a nonabelian Sylow $p$-subgroup of $G$ and $m(P)$ is the smallest non-linear irreducible character degree of $P$, we prove that there exists $χ\in {\rm Irr}(G)$ in the principal $p$-block of $G$ such that $1<χ(1)_p\le m(P)$, giving one inequality of the Eaton-Moretó conjecture for principal blocks. This, assuming Dade's Projective conjecture, implies the Eaton-Moretó conjecture for principal blocks.

• The paper proves that for a finite group with nonabelian Sylow p-subgroup P, the principal block contains an irreducible character χ with 1 < χ(1)ₚ ≤ pᵐ, establishing h ≤ m.
• The proof combines minimal-counterexample arguments, character-triple isomorphisms, strong McKay results, and Classification of Finite Simple Groups techniques, including separate analyses of Lie-type groups and the exceptional case (p, S) = (3, Co₃).
• Together with Eaton and Moretó’s result that Dade’s Projective Conjecture implies m ≤ h, the theorem shows that the full conjecture holds for principal blocks whenever Dade’s conjecture is assumed.

The paper proves one inequality of the Eaton–Moretó conjecture for principal blocks of finite groups, and shows that the full conjecture for principal blocks follows from Dade's Projective Conjecture. Specifically, if $G$ is a finite group, $p$ a prime, and $P$ a nonabelian Sylow $p$-subgroup of $G$, and $p^m = m(P)$ denotes the smallest non-linear irreducible character degree of $P$, then there exists $\chi \in \operatorname{Irr}(G)$ in the principal $p$-block $B_0(G)$ with

$p$0

This establishes the inequality $p$1 in the principal block case, where $p$2 is the smallest positive height among characters of the block. Combined with Eaton and Moretó's earlier reduction showing that Dade's Projective Conjecture implies $p$3, this yields: assuming Dade's Projective Conjecture holds for the principal block of $p$4, the Eaton–Moretó conjecture holds for principal blocks.

Background and context

Brauer's Height Zero Conjecture — now a theorem via work of Kessar–Malle, Malle–Navarro–Schaeffer Fry–Tiep, and Ruhstorfer — asserts that all irreducible characters in a block $p$5 have height zero exactly when the defect group is abelian. For nonabelian defect groups $p$6, Eaton and Moretó conjectured that the smallest positive height $p$7 among characters of $p$8 equals $p$9, where $P$0 is the smallest non-linear irreducible character degree of $P$1. Prior supporting evidence came from Brunat–Malle, Feng–Liu–Zhang, Malle–Moretó–Rizo, Malle–Schaeffer Fry, and Navarro's treatment of $P$2-solvable groups. Since Eaton and Moretó proved that Dade's Projective Conjecture implies $P$3, the remaining task was the reverse inequality, which is what this paper supplies for principal blocks (where the defect group is a Sylow $P$4-subgroup).

The proof relies on the Classification of Finite Simple Groups at several points, and on the recently completed strong forms of the McKay conjecture (Cabanes–Späth; Rossi), used through a character-triple isomorphism producing characters of controlled $P$5-part.

A key reduction theorem

The technical core is a block-free statement. Suppose $P$6 is a Sylow $P$7-subgroup of $P$8 and $P$9 is such that $p$0 is abelian and nonnormal in $p$1. Then there exists $p$2 with $p$3. The proof analyzes a minimal counterexample: after replacing $p$4 by a suitable intersection $p$5 of two distinct Sylow $p$6-subgroups (normal in both), the authors show such a counterexample has trivial $p$7, equals its own $p$8, and has a unique minimal normal subgroup $p$9, necessarily nonabelian with $G$0. A further lemma rules out the existence of any $G$1 whose stabilizer $G$2 is proper normal in $G$3 with abelian quotient of index at most $G$4 — otherwise Clifford theory and induction produce a forbidden character. This forces every "active" simple factor $G$5 of $G$6 (where $G$7) to be $G$8-invariant, reducing the problem to an almost simple group $G$9 possessing two distinct Sylow $p^m = m(P)$0-subgroups $p^m = m(P)$1 with $p^m = m(P)$2 normal in both and $p^m = m(P)$3.

Almost simple groups

The almost simple case is handled case by case. If $p^m = m(P)$4, a result derived from prior work yields a character in $p^m = m(P)$5, nontrivial on the socle, with $p^m = m(P)$6-part exactly $p^m = m(P)$7. If $p^m = m(P)$8 is simple and the Sylow $p^m = m(P)$9-subgroups are abelian, Zhang's defect group theorem combined with the "if" direction of Brauer's Height Zero Conjecture produces a character with $P$0 for a maximal intersection $P$1 of distinct Sylows containing $P$2. For nonabelian Sylows, known results give characters with $P$3 except when $P$4 or $P$5 is of Lie type in defining characteristic. The exceptional Conway group case is settled computationally: GAP calculation shows the minimal value of $P$6 over $P$7 is $P$8, and a check that $P$9 for every maximal subgroup $\chi \in \operatorname{Irr}(G)$0 of a Sylow 3-subgroup rules out $\chi \in \operatorname{Irr}(G)$1. For groups of Lie type in characteristic $\chi \in \operatorname{Irr}(G)$2, the argument identifies $\chi \in \operatorname{Irr}(G)$3 as $\chi \in \operatorname{Irr}(G)$4 over a nonempty union $\chi \in \operatorname{Irr}(G)$5 of Frobenius orbits of positive roots, hence a power $\chi \in \operatorname{Irr}(G)$6, while Brunat–Malle supply a principal-block character with $\chi \in \operatorname{Irr}(G)$7.

Proof of the main theorem

With the key theorem available, the main theorem follows by induction on $\chi \in \operatorname{Irr}(G)$8. One reduces to $\chi \in \operatorname{Irr}(G)$9 using block covering and degree preservation under quotients of $p$0-index. Abelian minimal normal subgroups are handled by a theorem showing that either the desired character exists or $p$1, allowing passage to $p$2; the argument here uses Brauer's Third Main Theorem to control which induced characters land in $p$3, together with a lemma guaranteeing a principal-block character lying over any $p$4 with $p$5. Nonabelian minimal normal subgroups $p$6 split into two cases: if $p$7 acts nontrivially on the factors, a construction using a nonprincipal $p$8-degree character of $p$9 invariant under $B_0(G)$0 produces a character with $B_0(G)$1; if $B_0(G)$2 acts trivially, then either some almost simple quotient $B_0(G)$3 has $B_0(G)$4 dividing $B_0(G)$5 (handled as above) or $B_0(G)$6 is a $B_0(G)$7-number, reducing to the simple group itself, where the Brunat–Malle theorem applies directly.

Limitations and open questions

Several dependencies should be noted plainly. The proof invokes the Classification of Finite Simple Groups multiple times, and depends on the recently proved McKay conjecture machinery (the inductive McKay condition); it therefore inherits whatever caveats attach to those results. The full Eaton–Moretó conjecture for arbitrary blocks remains open: the paper proves only the inequality $B_0(G)$8 for principal blocks, and the complementary inequality still rests on Dade's Projective Conjecture, which is unproved. The $B_0(G)$9 exception is handled by explicit computation rather than a uniform theoretical argument, and the Lie type in defining characteristic case requires a separate root-theoretic analysis rather than following from the general framework. Whether the methods extend from principal blocks to arbitrary blocks with nonabelian defect groups is not addressed.

Conclusion

The paper establishes that every finite group with nonabelian Sylow $p$00-subgroup $p$01 possesses a principal-block irreducible character whose degree has $p$02-part between $p$03 and $p$04, thereby proving half of the Eaton–Moretó conjecture in the principal block case and reducing the other half to Dade's Projective Conjecture. The proof combines a careful minimal-counterexample analysis built on McKay-type character triple isomorphisms with case-by-case verification over finite simple groups, including a computational check for $p$05 at $p$06.