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Finite groups whose maximal subgroups of order divisible by all the primes are supersolvable. (English) Zbl 1506.20029

Summary: We study finite groups \(G\) with the property that for any subgroup \(M\) maximal in \(G\) whose order is divisible by all the prime divisors of \(|G|\), \(M\) is supersolvable. We show that any nonabelian simple group can occur as a composition factor of such a group and that, if \(G\) is solvable, then the nilpotency length and the rank are arbitrarily large. On the other hand, for every prime \(p\), the \(p\)-length of such a group is at most 1. This answers questions proposed by V. Monakhov in The Kourovka Notebook.

MSC:

20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20E28 Maximal subgroups
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References:

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