On universal locally finite groups

We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2^ℵ ₀, and assuming G.C.H. for every cardinal whose confinality is >ℵ₀. However, if λ>κ when κ is strongly compact and of λ=ℵ₀, then there exists a universal locally finite group of cardinality λ. The idea is to use the failure of the amalgamation property in a strong sense. We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF. We would like to thank Simon Thomas for reading carefully a preliminary version of this paper, proving Lemma 20 and making valuable remarks. Also we thank the United States—Israel Binational Science Foundation for partially supporting this work.