Conclusion
Obviously we have not covered aU possible ascendant and descendant nilpotent properties. Using (1.1), (1.2), or other methods, and Lemma 1.2 it is possible to provide many more such properties. For exampleC q∼(Cpm×Cpm) is a metabelian group having infinitely many non-isomorphic Sylow-p-subgroups [6]. Thus, having one isomorphism class of Sylow-p-subgroups is a descendant nilpotent property.
The importance of such properties lies not only in their utility within the class of nilpotent groups but in the fact that they delineate, in the sense of (1.3) and (1.4), the class of nilpotent groups within the class of solvable groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
P. Hall, Finiteness conditions for soluble groups. Proc. London Math. Soc. (3)4, 419–436 (1954).
P. Hall, On the finiteness of certain soluble groups. Proc. London Math. Soc. (3)9, 595–622 (1959).
P. Hall, The Frattini subgroups of finitely generated groups. Proc. London Math. Soc. (3)11, 327–352 (1961).
D. J. S.Robinson, Finiteness conditions and generalized soluble groups. Berlin 1972.
R. B.Warfield, Nilpotent groups. LNM 513, Berlin 1976.
B. A. F. Wehrfritz, Sylow subgroups of locally finite groups with min-p. J. London Math. Soc. (2)1, 421–427 (1969).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fournelle, T.A. Properties that characterize classes of nilpotent groups. Arch. Math 41, 199–203 (1983). https://doi.org/10.1007/BF01194829
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01194829