Amalgamated products of profinite groups: Counterexamples

Ribes, Luis

doi:10.1090/S0002-9939-1973-0382463-7

Amalgamated products of profinite groups: Counterexamples
HTML articles powered by AMS MathViewer

by Luis Ribes PDF

Proc. Amer. Math. Soc. 37 (1973), 413-416 Request permission

Abstract:

If ${A_1}$ and ${A_2}$ are profinite groups with a common closed subgroup H, the profinite amalgamated product of ${A_1}$ and ${A_2}$ over H is said to exist if ${A_1}$ and ${A_2}$ are canonically embedded in the push-out of ${A_1}$ and ${A_2}$ over H in the category of profinite groups. It is proved that such products need not exist.

References

Gilbert Baumslag, On the residual finiteness of generalised free products of nilpotent groups, Trans. Amer. Math. Soc. 106 (1963), 193–209. MR 144949, DOI 10.1090/S0002-9947-1963-0144949-8
K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29–62. MR 87652, DOI 10.1112/plms/s3-7.1.29
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
Graham Higman, A remark on finitely generated nilpotent groups, Proc. Amer. Math. Soc. 6 (1955), 284–285. MR 69176, DOI 10.1090/S0002-9939-1955-0069176-5
Barry Mitchell, Theory of categories, Pure and Applied Mathematics, Vol. XVII, Academic Press, New York-London, 1965. MR 0202787
Luis Ribes, Introduction to profinite groups and Galois cohomology, Queen’s Papers in Pure and Applied Mathematics, No. 24, Queen’s University, Kingston, Ont., 1970. MR 0260875
Luis Ribes, On amalgamated products of profinite groups, Math. Z. 123 (1971), 357–364. MR 291295, DOI 10.1007/BF01109991

Productos amalgamados de grupos pronilpotentes

Jean-Pierre Serre, Cohomologie galoisienne, Lecture Notes in Mathematics, No. 5, Springer-Verlag, Berlin-New York, 1965 (French). With a contribution by Jean-Louis Verdier; Troisième édition, 1965. MR 0201444