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Remarks on the Upper Centralc Series of a Group

Published online by Cambridge University Press:  18 May 2009

D. H. McLain
D. H. McLain
Affiliation:
The University Glasgow
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Following, for example, Kurošs [8], we define the (transfinite) upper central series of a group G to be the series

such that Zα + 1/Za is the centre of G/Zα, and if β is a limit ordinal, then If α is the least ordinal for which Zα =Zα+1=…, then we say that the upper central series has length α, and that Zα= His the hypercentre of G. As usual, we call G nilpotent if Zn= Gfor some finite n.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 3 , Issue 1 , December 1956 , pp. 38 - 44
Copyright
Copyright © Glasgow Mathematical Journal Trust 1956

References

REFERENCES

1.Baer, R., Das Hyperzentrum einer Gruppe III, Math. Z., 59 (1953–1954), 296–338.CrossRefGoogle Scholar
2.Baer, R., Supersoluble groups, Proc. Amer. Math. Soc., 6 (1955), 16–32.CrossRefGoogle Scholar
3.Birkhoff, G., Lattice theory (Amer. Math. Soc, 1948).Google Scholar
4.Duguid, A. M. and McLain, D. H., FC-soluble and FC-nilpotent groups, Proc. Comb. Phil.Soc., 52 (1956), 391–398.CrossRefGoogle Scholar
5.Haimo, F., On the FC-chain of a group, Canad. Jour. Math., 5 (1953), 498–511.CrossRefGoogle Scholar
6.Hall, P., A contribution to the theory of groups of prime power order, Proc. Lond. Math.Soc., 36 (1933), 29–95.Google Scholar
7.Hall, P., Finiteness conditions for soluble groups, Proc. Lond. Math. Soc., (3) 4 (1954), 419–436.Google Scholar
8.Kurošs, A. G., Theory of groups, (Chelsean, 1955).Google Scholar
9.Mal'cev, A. I., Generalised nilpotent algebras and their associated groups, Mat. Sbornik N.S., 25 (1949), 347–366. (in Russian).Google Scholar
10.McLain, D. H., On locally nilpotent groups, Proc. Camb. Phil. Soc., 52 (1956), 5–11.CrossRefGoogle Scholar
11.Weir, A. J., The Sylow p-subgroups of the general linear group over finite fields of characteristic p, Proc. Amer. Math. Soc., 6 (1955), 454–464.Google Scholar