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Free products and residual nilpotency

Published online by Cambridge University Press:  17 April 2009

I. M. S. Dey
I. M. S. Dey
Affiliation:
Department of Pure Mathematics, SGS, Australian National University, Canberra. A.C.T.
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Abstract

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Let G be the free product of groups Gα, [Gα] the cartesian subgroup of G and k [Gα] the intersection of [G] with the k–th term of the lower central series for G. Then the k[Gα] form a descending chain of subgroups of ] and it is shown that if the intersection of all the subgroups in this chain is trivial then G and hence each Gα, is residually nilpotent. This answers a question of S. Moran.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 1 , Issue 1 , April 1969 , pp. 11 - 13
Copyright
Copyright © Australian Mathematical Society 1969

References

[1] Moran, S., “Associative operations on groups II” Proc. London Math. Soc. (3) 8 (1958), 548–568.CrossRefGoogle Scholar
[2] Baer, Reinhold und Levi, Friedrich, “Freie Produkte und ihre Untergruppen”, Compositio Math. 3 (1936), 391–398.Google Scholar
[3] Dey, Ian M.S., “Relations between the free and direct products of groups”,.Math. Zeitsahr. 80 (1962), 121–147.CrossRefGoogle Scholar
[4] MacLane, Saunders, “A proof of the subgroup theorem for free products”, Mathematika 5 (1958), 13–19.CrossRefGoogle Scholar