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19.55 (2018)

Solved

Suppose that in a finite group $G$ every maximal subgroup $M$ is supersoluble whenever $\pi(M) = \pi(G)$, where $\pi(G)$ is the set of all prime divisors of the order of $G$.
$\qquad$ a) What are the non-abelian composition factors of $G$?
$\qquad$ b) Determine the exact upper bounds for the nilpotency length, the rank, and the $p$-length of $G$ if $G$ is soluble.

Contributor: V. S. Monakhov

Progress

a) Every nonabelian finite simple group can occur as a composition factor of $G$ (A. Moretó, Monatsh. Math., 195, no. 3 (2021), 497–500).

b) There is not any bound for the nilpotency length or the rank, but the $p$-length is at most 1 for every prime $p$ (A. Moretó, Monatsh. Math., 195, no. 3 (2021), 497–500).

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