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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A generalisation of supersoluble groups


Author: R. J. Haggarty
Journal: Trans. Amer. Math. Soc. 209 (1975), 433-441
MSC: Primary 20D10
MathSciNet review: 0372023
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Abstract: A $ p$-soluble group $ G$ belongs to the class $ F(n,p)$ whenever the ranks of the $ p$-chief factors of $ G$ divide $ n$ and $ G$ has order coprime to $ n$. A group in $ F(n,p)$ is characterised by the embedding of its maximal subgroups. Whenever $ {N_1}$ and $ {N_2}$ are normal subgroups of $ G$, of coprime indices in $ G$, which lie in $ F(n,p)$, then $ G$ lies in $ F(n,p)$ also. $ F(n)$ denotes the intersection, taken over all primes $ p$, of the classes $ F(n,p)$. Simple groups all of whose proper subgroups lie in $ F(n)$ are determined. Given an integer $ n > 2$, there exist an integer $ m$ with the same prime divisors as $ n$ and a soluble group $ G$ such that $ G$ lies in $ F(m)$ but $ G$ does not possess a Sylow tower. (We may take $ m = n$ provided that $ n$ is not a multiple of 1806.) Furthermore, when $ n$ is odd, an example of a soluble group $ G$, all of whose proper subgroups lie in $ F(n)$ but $ G$ has no Sylow tower, is given.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0372023-2
Keywords: Supersoluble groups, saturated formations, Fitting class, Sylow tower
Article copyright: © Copyright 1975 American Mathematical Society