Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T06:58:07.608Z Has data issue: false hasContentIssue false
  • English
  • Français

A converse to Lagrange's theorem

Published online by Cambridge University Press:  09 April 2009

T. R. Berger
T. R. Berger
Affiliation:
Department of Mathematics, University of Minnesota, Minneapolis, Minn., U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An integer n >0 is called a CLT-number if any group of order n has subgroups of order d for every divisor d of n. The Set of CLT-numbers n is characterized by properties of the prime factorization of n. In addition, if G has order dividing a CLT-number then the structure of G/CG(U) is given where U is a chief factor of G. As a consequence, it is shown that G is solvable of Fitting height at most 3.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 25 , Issue 3 , May 1978 , pp. 291 - 313
Copyright
Copyright © Australian Mathematical Society 1978

References

Berger, T. R. (1973a), ‘Hall–Higman type theorems. IV’, Proc. Amer. Math. Soc. 37, 317–325.CrossRefGoogle Scholar
Thomas, R. Berger (1973b), ‘Nilpotent fixed point free automorphism groups of solvable groups’, Math. Z. 131, 305–312.Google Scholar
Berger, T. R. (1975), ‘Hall–Higman type theorems. II’, Trans. Amer. Math. Soc. 205, 47–69.CrossRefGoogle Scholar
Henry, G. Bray (1968), ‘A note on CLT groups’, Pacific J. Math. 27, 229–231.Google Scholar
Deskins, W. E. (1968), ‘A characterization of finite supersolvable groups’, Amer. Math. Monthly 75, 180–182.CrossRefGoogle Scholar
Klaus, Doerk (1966), ‘Minimal nicht überauflösbare, endliche Gruppen’, Math. Z. 91, 198–205.Google Scholar
Walter, Feit and John, G. Thompson (1963), ‘Solvability of groups of odd order’, Pacific J. Math. 13, 775–1029.Google Scholar
Gagen, T. M. (to appear), ‘A note on groups with the inverse Lagrange property’, Proc. Miniconf. Group Theory, Canberra 1975 (Lecture Notes in Mathematics).Google Scholar
Daniel, Gorenstein (1968), Finite Groups (Harper's Series in Modern Mathematics. Harper and Row, New York, Evanston, London, 1968).Google Scholar
Huppert, B. (1967), Endliche Gruppen I (Die Grundlehren der mathematischen Wissenschaften, 134. Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
Donald, J. McCarthy (1971), ‘A survey of partial converses to Lagrange's theorem on finite groups’, Trans. New York Acad. Sci. (2) 33, 586–594.Google Scholar
McLain, D. H. (1957), ‘The existence of subgroups of given order in finite groups’, Proc. Cambridge Philos. Soc. 53, 278–285.CrossRefGoogle Scholar
Oystein, Ore (1939), ‘Contributions to the theory of groups of finite order’, Duke Math. J. 5, 431–460.Google Scholar
Donald, Passman (1968), Permutation Groups (Mathematics Lecture Note Series. Benjamin, New York, Amsterdam, 1968).Google Scholar
Struik, R. R. (preprint), ‘Partial converses to Lagrange's theorem’.Google Scholar
Thompson, J. G. (1974), ‘Simple 3'-groups’, Symposia Mathematica, 8, 517–530 (Convegno di Gruppi e loro Rappresentazioni, INDAM, Roma, 1972. Academic Press, London, 1974).Google Scholar
Weir, A. J. (1955), ‘Sylow ρ-subgroups of the classical groups over finite fields with characteristic prime to ρ’, Proc. Amer. Math. Soc. 6, 529–533.Google Scholar
David, L. Winter (1972), ‘The automorphism group of an extraspecial ρ-group’, Rocky Mountain J. Math. 2, 159–168.Google Scholar
Guido, Zappa (1940), ‘Remark on a recent paper of O. Ore’, Duke Math. J. 6, 511–512.Google Scholar