Complete universal locally finite groups

Abstract:

This paper will partly strengthen a recent application of model theory to the construction of sets of pairwise nonembeddable universal locally finite groups [8]. Our result is Theorem. There is a set $\mathcal {U}$ of ${2^{{\aleph _1}}}$ universal locally finite groups of order ${\aleph _1}$ with the following properties: 0.1. If $U \ne V \in \mathcal {U}$ and A and B are uncountable sugroups of U and V, then A and B are not isomorphic. Let A be an uncountable subgroup of $U \in \mathcal {U}$. 0.2. A does not belong to any proper variety of groups, and 0.3. A is not isomorphic to any of its proper subgroups. 0.4. Every $U \in \mathcal {U}$ is a complete group (every automorphism of U is inner).