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Kronecker classes of fields and covering subgroups of finite groups

Part of: Field extensions Permutation groups

Published online by Cambridge University Press:  09 April 2009

Cheryl E. Praeger
Cheryl E. Praeger
Affiliation:
University of Western Australia, Nedlands, WA 6009, Australia
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Abstract

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Kronecker classes of algebraci number fields were introduced by W. Jehne in an attempt to understand the extent to which the structure of an extension K: k of algebraic number fields was influenced by the decomposition of primes of k over K. He found an important link between Kronecker equivalent field extensions and a certain covering property of their Galois groups. This surveys recent contributions of Group Theory to the understanding of Kronecker equivalence of algebraic number fields. In particular some group theoretic conjectures related to the Kronecker class of an extension of bounded degree are explored.

MSC classification

Secondary: 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures 12F10: Separable extensions, Galois theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 57 , Issue 1 , August 1994 , pp. 17 - 34
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Bauer, M., ‘Zur Theorie der algebraischen Zahlkörper’, Math. Ann. 77 (1916), 353–356.CrossRefGoogle Scholar
[2]Borovik, A., ‘The structure of finite subgroups of simple algebraic groups’, Algebra and Logic (3) 28 (1989), 249–279 (in Russian).Google Scholar
[3]Brandl, R., Eine Überdeckungseigenschaft endlicher Gruppen (Diplomarbeit, Würzburg, 1976).Google Scholar
[4]Brandl, R., ‘A covering property of finite groups’, Bull. Austral. Math. Soc. 23 (1981), 227–235.CrossRefGoogle Scholar
[5]Caranti, A., Gavioli, N. and Mattarei, S., ‘Subgroups of finite p-groups inducing the same permutation character’, Comm. in Algebra, to appear.Google Scholar
[6]Fein, B., Kantor, W. M. and Schacher, M., ‘Relative Brauer groups, II’, J. reine angew. Math. 238 (1981), 39–57.Google Scholar
[7]Feit, W. and Seitz, G. M., ‘On finite rational groups and related topics’, Illinois J. Math. 33 (1988), 103–131.Google Scholar
[8]Gassmann, F., ‘Bemerkungen zur vorstehenden Arbeit von Hurwitz’, Math. Z. 25 (1926), 665–675.Google Scholar
[9]Guralnick, R. M., ‘Zeros of permutation characters with applications to prime splitting and Brauer groups’, J. Algebra 131 (1990), 294–302.Google Scholar
[10]Guralnick, R. M., Liebeck, M. W. and Praeger, C. F., ‘Covering subgroups and arithmetic properties of number fields’, in preparation.Google Scholar
[11]Guralnick, R. M. and Saxl, J., ‘Primitive permutation characters’, Groups, combinatorics and geometry (eds. Liebeck, M. W. and Saxl, J.), London Math. Soc. Lecture Notes Ser. 165 (Cambridge Univ. Press, Cambridge, 1992) pp. 364–367.CrossRefGoogle Scholar
[12]Guralnick, R. M. and Stern, L., ‘Solitary Galois extensions of algebraic number fields’, J. Number Theory, to appear.Google Scholar
[13]Jehne, W., ‘Kronecker classes of algebraic number fields’, J. Number Theory 9 (1977), 279–320.CrossRefGoogle Scholar
[14]Jehne, W., ‘On Kronecker classes of atomic extensions’, Proc. London Math. Soc. (3) 34 (1977), 32–64.Google Scholar
[15]Klingen, N., ‘Zahlkörper mit gleicher Primzerlegung’, J. reine angew. Math. 299 (1978), 342–384.Google Scholar
[16]Klingen, N., ‘Atomare Kronecker-Klassen mit speziellen Galoisgruppen’, Abh. Math. Sem. Univ. Hamburg. 48 (1979), 42–53.CrossRefGoogle Scholar
[17]Klingen, N., ‘Uber schwache quadratische Zerlegungsgestze’, Comment. Math. Helv. 55 (1980), 645–651.Google Scholar
[18]Klingen, N., ‘Rigidity of decomposition laws and number fields’, J. Austral. Math. Soc. (Seies A) 51 (1991), 171–186.CrossRefGoogle Scholar
[19]‘The Kourovka Notebook’, (Mathematics Institute of the Siberian Division of the Academy of Sciences of the USSR, Novosibirsk, 1990).Google Scholar
[20]Kronecker, L., ‘Über die Irreductibilität von Gleichungen’, Werke, II, 85–93, Monatsber. Deut. Akad. Wiss, (1880), 155–163.Google Scholar
[21]Lang, S., Algebraic Number Theory (Springer-Verlag, New York, 1986).Google Scholar
[22]Liebeck, M. W., Praeger, C. F. and Saxl, J., ‘On the O'Nan-Scott Theorem for finite primitive permutation groups’, J. Austral. Math. Soc. (Series A) 44 (1988), 389–396.CrossRefGoogle Scholar
[23]Lochter, M., Neue zahlentheoretische Aspekte der Kronecker-Äquivalenz (Doctoral thesis, University of Köln, Köln, 1992).Google Scholar
[24]Perlis, R., ‘On the equation ζk(s) = ζk'(s)’, J. Number Theory 9 (1977), 342–360.Google Scholar
[25]Praeger, C. E., ‘Covering subgroups of groups and Kronecker classes of fields’, J. Algebra 118 (1988), 455–463.CrossRefGoogle Scholar
[26]Praeger, C. E., ‘On octic extensions and a problem in group theory’, in: Group Theory, Proceedings of the 1987 Singapore Group Theory Conference (eds. Cheng, K. N. and Leong, Y. K.) (De Gruyter, Berlin, 1989) pp. 443–463.Google Scholar
[27]Praeger, C. E., ‘Kronecker classes of field extensions of small degree’, J. Austral. Math. Soc. (Series A) 50 (1991), 297–315.CrossRefGoogle Scholar
[28]Pyber, L., ‘Finite groups have many conjugacy classes’, J. London Math. Soc. (2) 46 (1992), 239–249.CrossRefGoogle Scholar
[29]Saxl, J., ‘On a question of W. Jehne concerning covering subgroups of groups and Kronecker classes of fields’, J. London Math. Soc. (2) 38 (1988), 243–249.CrossRefGoogle Scholar
[30]Stern, L., ‘On the equality of norm groups of global fields’, J. Number Theory 36 (1990), 108–126.Google Scholar