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On group actions on groups and associated series

Published online by Cambridge University Press:  24 October 2008

Peter Hilton
Affiliation:
Battelle Research Centre, Seattle and E.T.H., Zürich
Urs Stammbach
Affiliation:
Battelle Research Centre, Seattle and E.T.H., Zürich

Extract

1. Introduction. In this paper we consider Q-groups; that is, Q is a group and we consider groups N endowed with a Q-action, meaning a homomorphism of Q into the group of automorphisms of N. In (4) a lower central Q-series was defined for such a Q-group, N, generalizing the lower central series of a group, and results were obtained relating to the localization of such a series. Since the ideas in that paper were inspired by the homotopical localization theory of nilpotent spaces (see (6)), the main body of results in (4) was concerned with the case in which N is nilpotent, and perhaps also the group Q and the action of Q on N (in the sense that the lower central Q-series terminates after a finite number of steps with the trivial group {1}). We now adopt a broader view-point and only restrict ourselves to the nilpotent case when our results appear to require us to do so; thus the spirit of this paper is much more that of general group theory as presented in [(8), especially ch. VI]. Thus, while there is some overlap of results, the methods used are not the same and many results (for example, Theorem 3·1) are far more general than any obtained in (4). Moreover, the methods also appear to us to be more appropriate in that essential appeal was made in (4) to a sophisticated theorem of Norman Blackburn on nilpotent groups, whereas here we merely use homological methods, the construction of the semidirect product and, in section 4, some very classical facts of the commutator calculus.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1976

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References

REFERENCES

(1)Hall, P. and Hartley, B.The stability group of a series of subgroups. Proc. London Math. Soc. 16 (1966), 139.CrossRefGoogle Scholar
(2)Hilton, P.Localization and cohomology of nilpotent groups. Math. Z. 132 (1973), 263286.CrossRefGoogle Scholar
(3)Hilton, P. On direct limits of nilpotent groups. Localization in homotopy theory and related topics, Springer Lecture Notes 418 (1974), 6877.CrossRefGoogle Scholar
(4)Hilton, P. Nilpotent actions on nilpotent groups. Conference on algebra and logic. Springer Lecture Notes 450 (1975), 174196.Google Scholar
(5)Hilton, P. and Mislin, G.Bicartesian squares of nilpotent groups. Comment. Math. Heiv. (to appear).Google Scholar
(6)Hilton, P., Mislin, G. and Roitberg, J.Localization of nilpotent groups and spaces. Mathematics Studies, Vol. 15 (North-Holland, 1975).CrossRefGoogle Scholar
(7)Hilton, P. and Stammbach, U.Localization and isolators. Houston J. Math. (to appear).Google Scholar
(8)Stammbach, U.Homology in group theory. Springer Lecture Notes 359 (1973).CrossRefGoogle Scholar