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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 $p>q$, $p\mid |G:H|$, $q\mid |G:K|$, then $p¬\equiv 1\left(q\right)$, and each is quasinormal in the other; (4) ${G}^{\text{'}}$ 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

##### MSC:
 20D40 Products of subgroups of finite groups 20D10 Solvable finite groups, theory of formations etc. 20D20 Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure
##### References:
 [1] R. Baer, Classes of finite groups and their properties. Illinois J. Math.1, 115-187 (1957). [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 [3] O. H. Kegel, Zur Struktur mehrfach faktorisierbarer endlicher Gruppen. Math. Z.87, 42-48 (1965). · Zbl 0123.02503 · doi:10.1007/BF01109929 [4] D.Gorenstein, Finite groups. New York 1968. [5] K. Doerk, Minimal nicht überauflösbare, endliche Gruppen. Math. Z.91, 198-205 (1966). · Zbl 0135.05401 · doi:10.1007/BF01312426 [6] M.Hall, The theory of groups. New York 1959. [7] B.Huppert, Endliche Gruppen I. Berlin-Heidelberg-New York 1967. [8] W. R.Scott, Group theory. Englewood Cliffs, New Jersey 1964. [9] B. Huppert, Monomiale Darstellung endlicher Gruppen. Nagoya Math. J.6, 93-94 (1953).