Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-24T13:03:38.561Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on commutators in free products

Published online by Cambridge University Press:  24 October 2008

H. B. Griffiths
H. B. Griffiths
Affiliation:
King's College†Aberdeen, Scotland
Get access Rights & Permissions [Opens in a new window]

Extract

In (1), Higman introduces the unrestricted free product of a set of groups, and gives it a natural topology. When this set is an infinite sequence of free cyclic groups he denotes by F their unrestricted free product; we shall denote by K the product for a general sequence {Gn} and, throughout the paper, we assume that each Gn is non-trivial and countable. Higman proves as an incidental result that the commutator subgroup [F, F] is not closed in the topological group F, and the first object of this note is to generalize this result to K. From this, the more interesting deduction immediately follows that K is never equal to L = [K, K]. Indeed, we prove in fact that the cardinal of L and its index in K are both c, the cardinal of the continuum.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 178 - 188
Copyright
Copyright © Cambridge Philosophical Society 1954

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Higman, G.Unrestricted free products, and varieties of topological groups. J. Lond. math. Soc. 27 (1952), 73–81.CrossRefGoogle Scholar
(2)Magnus, W.Beziehungen zwischen Gruppen und Idealen in einen speziellen Ring. Math. Ann. 111 (1935), 259–80.CrossRefGoogle Scholar
(3)Van der Waerden, B. L.Free products of groups. Amer. J. Math. 70 (1948), 527–8.CrossRefGoogle Scholar