zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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

MSC:
20D40Products of subgroups of finite groups
20D10Solvable finite groups, theory of formations etc.
20D20Sylow subgroups of finite groups, Sylow properties, π-groups, π-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).