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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A generalisation of supersoluble groups
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by R. J. Haggarty PDF
Trans. Amer. Math. Soc. 209 (1975), 433-441 Request permission

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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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 209 (1975), 433-441
  • MSC: Primary 20D10
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0372023-2
  • MathSciNet review: 0372023