Isaacs–Seitz conjecture on solvable-group derived length

From papers

For a finite group GG, let Irr(G)\operatorname{Irr}(G) be the set of irreducible characters and define the set of irreducible character degrees by

cd(G)=χ(1)χIrr(G).\operatorname{cd}(G)=\\{\chi(1)\mid\chi\in\operatorname{Irr}(G)\\}.

Let dl(G)\operatorname{dl}(G) denote the derived length of GG. Isaacs–Seitz conjecture. If GG is solvable, then

dl(G)cd(G).\operatorname{dl}(G)\le |\operatorname{cd}(G)|.

The conjecture has been proved in many cases, but the general problem remains open; the source records Gluck's weaker bound dl(G)2cd(G)\operatorname{dl}(G)\le2|\operatorname{cd}(G)|.

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Sources & referencesView supporting material

Primary source

Alexander Moretó, “The Main Problem of Block Theory: Picky Elements and Subnormalizers”, arXiv:2604.24565 (2026).

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