18.102 (2014)
Open(E. C. Dade). Let $C$ be a Carter subgroup of a finite solvable group $G$, and let $\ell(C)$ be the number of primes dividing $|C|$ counting multiplicities. It was proved in (E. C. Dade, Illinois J. Math., 13 (1969), 449–514) that there is an exponential function $f$ such that the nilpotent length of $G$ is at most $f(\ell(C))$. Is there a linear (or at least a polynomial) function $f$ with this property?
Progress
Editors’ comment: A quadratic function with that property was proved to exist in the special case where $G = H \rtimes C$ and $C$ is a cyclic subgroup such that $C_H(C) = 1$, (E. Jabara, J. Algebra, 487 (2017), 161–172).
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