Carter subgroups and Fitting heights of finite groups

Abstract

Let G be a finite group possessing a Carter subgroup K. Denote by $$\mathbf {h}(G)$$ the Fitting height of G, by $$\mathbf {h}^*(G)$$ the generalized Fitting height of G, and by $$\ell (K)$$ the number of composition factors of K, that is, the number of prime divisors of the order of K with multiplicities. In 1969, E. C. Dade proved that if G is solvable, then $$\mathbf {h}(G)$$ is bounded in terms of $$\ell (K)$$ . In this paper, we show that $$\mathbf {h}^*(G)$$ is bounded in terms of $$\ell (K)$$ as well.