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Zbl 0685.20018
On the supersolvability of finite groups.
(English)
[J] Arch. Math. 53, No. 4, 318-326 (1989). ISSN 0003-889X; ISSN 1420-8938/e

The object of this paper is to find sufficient conditions for the finite group $G=HK$, the product of two subgroups, to be supersolvable. The main sets of conditions are: (1) $H$ and $K$ are supersolvable and each subgroup of $H$ is quasinormal in $K$ ($H$ is quasinormal in $K$ if $HL=LH$ for all subgroups $L$ of $K$); (2) $H$ is nilpotent, $K$ is supersolvable and each is quasinormal in the other; (3) $H$ and $K$ are supersolvable, have coprime indices, for each pair of primes $p,q$ with $p>q$, $p\mid|G:H|$, $q\mid|G:K|$, then $p\not\equiv 1(q)$, and each is quasinormal in the other; (4) $G'$ is nilpotent and each of $H,K$ is supersolvable and quasinormal in the other. These results generalize work of {\it R. Baer} [Ill. J. Math. 1, 115-187 (1957; Zbl 0077.03003)], {\it D. K. Friesen} [Proc. Am. Math. Soc. 30, 46-48 (1971; Zbl 0232.20037)] and {\it O. H. Kegel} [Math. Z. 87, 42-48 (1965; Zbl 0123.02503)].
[J.D.P.Meldrum]
MSC 2000:
*20D40 Products of subgroups of finite groups
20D10 Solvable finite groups
20D20 Sylow subgroups of finite groups

Keywords: Sylow towers; products of subgroups; finite groups; supersolvable; quasinormal; nilpotent

Citations: Zbl 0077.03003; Zbl 0232.20037; Zbl 0123.02503

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