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On the supersolvability of finite groups. (English) Zbl 0685.20018
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|G:H|, q|G:K|, then p¬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 R. Baer [Ill. J. Math. 1, 115-187 (1957; Zbl 0077.03003)], D. K. Friesen [Proc. Am. Math. Soc. 30, 46-48 (1971; Zbl 0232.20037)] and O. H. Kegel [Math. Z. 87, 42-48 (1965; Zbl 0123.02503)].
Reviewer: J.D.P.Meldrum

20D40Products of subgroups of finite groups
20D10Solvable finite groups, theory of formations etc.
20D20Sylow subgroups of finite groups, Sylow properties, π-groups, π-structure
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[2]D. R. Friesen, Products of normal supersolvable subgroups. Proc. Amer. Math. Soc.30, 46-48 (1971). · doi:10.1090/S0002-9939-1971-0280590-4
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