## 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 $$p\not\equiv 1(q)$$, 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 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 abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure

### Citations:

Zbl 0077.03003; Zbl 0232.20037; Zbl 0123.02503
Full Text:

### References:

 [1] R. Baer, Classes of finite groups and their properties. Illinois J. Math.1, 115-187 (1957). · Zbl 0077.03003 [2] D. R. Friesen, Products of normal supersolvable subgroups. Proc. Amer. Math. Soc.30, 46-48 (1971). · Zbl 0232.20037 [3] O. H. Kegel, Zur Struktur mehrfach faktorisierbarer endlicher Gruppen. Math. Z.87, 42-48 (1965). · Zbl 0123.02503 [4] D.Gorenstein, Finite groups. New York 1968. · Zbl 0185.05701 [5] K. Doerk, Minimal nicht überauflösbare, endliche Gruppen. Math. Z.91, 198-205 (1966). · Zbl 0135.05401 [6] M.Hall, The theory of groups. New York 1959. · Zbl 0084.02202 [7] B.Huppert, Endliche Gruppen I. Berlin-Heidelberg-New York 1967. · Zbl 0217.07201 [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). · Zbl 0053.01201
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