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On Groups with Nilpotent Commutator Subgroups

Published online by Cambridge University Press:  22 January 2016

Nobuo Inagaki
Nobuo Inagaki*
Affiliation:
Tokyo Electrical Engineering College
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It is well known that a supersoluble group has the nilpotent commutator subgroup. But the converse does not hold in general. Therefore, it will be interesting to study the class of groups with nilpotent commutator subgroups. Some results were obtained by B. Huppert [1] and R. Baer [3] on this subject.

It is easily seen that the p-length of a group with the nilpotent commutator subgroup is 1 for every p dividing the order of this group. Theorem 1 of this paper may be considered as its local refinement, and is proved in a similar way as B. Huppert [2]. Theorem 2 gives a necessary and sufficient condition for a group to have the nilpotent commutator subgroup in terms of its Hall subgroups. In the last theorem 3 we shall study an interesting class of groups which have maximal subgroups of a certain nature. It turns out that these groups have nilpotent commutator subgroups and admit Sylowtowers.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 25 , 1965 , pp. 205 - 210
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1965

References

[1] Huppert, B., Normalteiler und maximale Untergruppen endlicher Gruppen. Math. Zeit. 60, 409–409 (1954).Google Scholar
[2] Huppert, B., Zur Sylowstruktur auflõsbarer Gruppen. Arch. Math. Vol. XII, 161–161 (1961).Google Scholar
[3] Baer, R., Classes of finite groups and their properties. Illinois J. Vol. 1, 115–115 (1957).Google Scholar
[4] Gaschütz, W., über die Ø-Untergruppen endlicher Gruppen. Math. Zeit. 58, 160–160 (1953).Google Scholar
[5] Hattori, A., On certain characteristic subgroups of a finite group. Jour, of the Math. Soci, of Japan, Vol. 13, 85–85 (1961).Google Scholar
[6] Ito, N., Über das Produkt von zwei abelschen Gruppen. Math. Zeit. 64, 400–400 (1955).Google Scholar
[7] Taunt, D. R., On A-groups. Proc. Cambridge Phil. Soc. 45, 24–24 (1949).Google Scholar
[8] Hall, P., and Higman, G., On the ¿-length of ¿-soluble group. Proc. London Math. Soc. 6, 1–1 (1956).Google Scholar
[9] Thompson, J., Finite groups with fixed-point-free automorphisms of prime order. Proc. Nat. Acad. Sci. 45, 578–578 (1959).Google Scholar
[10] Ishii, T., and Morita, I., On the Ø1-group and Ø2-group. Sugaku. Vol. 14, 41–41 (1963) (in Japanese).Google Scholar
[11] Zassenhaus, H., Lehrbuch der Gruppentheorie I. (1937).Google Scholar