Abstract
A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2a 3b and to a list of known “almost simple” groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. I. Alperin, R. Brauer and D. Gorenstein,Finite simple groups of 2-rank two, Scripta Math.29 (1977), 191–214.
J. L. Alperin, R. Brauer and D. Gorenstein,Finite groups with quasidihedral and wreathed Sylow 2-subgroups, Trans. Amer. Math. Soc.151 (1970), 1–261.
D. M. Bloom,The subgroups of PSL (3,q)for q odd, Trans. Amer. Math. Soc.127 (1967), 150–178.
R. Brauer and M. Suzuki,On finite groups of even order whose 2-Sylow subgroup is a quaternion group, Proc. Nat. Acad. Sci. U.S.A.45 (1959), 1757–1759.
W. Feit,The current situation in the theory of finite simple groups, Actes Congrès Intern. Math. 1970, Tome 1, pp. 55–93.
B. Gordon and M. Schacher,The admissibility of A 5, J. Number Theory11 (1979), 498–504.
B. Gordon and M. Schacher,Quartic coverings of a cubic, J. Number Theory, to appear.
D. Gorenstein,Finite Groups, Harper and Row, New York, 1968.
K.W. Gruenberg,Cohomological Topics in Group Theory, Lecture Notes, Springer-Verlag, 1970.
B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin, 1967.
I. M. Isaacs,Character Theory of Finite Groups, Academic Press, New York, 1976.
G. Janusz,Algebraic Number Fields, Academic Press, 1973.
J. Neukirch,On solvable number fields, Invent. Math.53 (1979), 135–164.
M. Schacher,Subfields of division rings I, J. Algebra9 (1968), 451–477.
I. Schur,Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math.132 (1907), 85–137.
I. Schur,Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen, J. Reine Angew. Math.139 (1918), 155–250.
J. Sonn, SL (2,5)and Frobenius Galois groups over Q, Canad. J. Math.32 (1980), 281–293.
J. Sonn,Rational division algebras as solvable crossed products, Israel J. Math.37 (1980), 246–250.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chillag, D., Sonn, J. Sylow-metacyclic groups andQ-admissibility. Israel J. Math. 40, 307–323 (1981). https://doi.org/10.1007/BF02761371
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02761371