Amalgamated Products of Profinite Groups: Counterexamples
Luis Ribes · Proceedings of the American Mathematical Society · 1973
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.