Complete Universal Locally Finite Groups of Large Cardinality

Abstract

This chapter discusses the complete universal locally finite (ULF) groups of large cardinality. A group G is in the class ULF of universal locally finite groups if (1) G is locally finite, (2) every finite group is isomorphic to a subgroup of G, and (3) any two isomorphic finite subgroups of G are conjugate in G. There is a countable ULF group C that is unique up to isomorphism. Every locally finite group of cardinality x is contained in a ULF group of cardinality x. Every countable locally finite group is embedded in C. Hickin's results are partially extended to arbitrary successor cardinals. Macintyre and Shelah used Ehrenfeucht–Mostowski models to construct their nonisomorphic ULF groups. The methods of Hickin and Macintyre and Shelah rely heavily on the fact that the free product with an amalgamation of two finite groups is residually finite. The chapter discusses the constricted symmetric group and proves the technical lemmas that form the heart of the construction.