Partial homogeneity in locally finite groups (Q1077531)

The main result herein is the construction of countably infinite locally finite simple groups G that satisfy a partial homogeneity condition on the finite subgroups of G. Specifically, if \(\pi\) is a non-empty set of primes and \(K=\{k_ p|\) \(p\in \pi '\}\) is a set of positive integers, there is a group G satisfying each of the following three conditions: (1) if \(\Omega\) is a ''classical family of finite groups'' (i.e., symmetric, alternating, SL, PSL, etc.) then G is the union of \(\Omega\)-groups; (2) isomorphic finite \(\pi\)-subgroups of G are conjugate in G; (3) if \(p\in \pi '\) and \(n\geq 1\), the number of G-conjugacy classes of elements of order \(p^ n\) equals \(k_ p+1\) if \(n\geq k_ p\) and equals \(n+1\) if \(n<k_ p.\) A byproduct of this construction is the display of \(2^{\aleph_ 0}\)- isomorphism types of countable direct limits of \(\Omega\)-groups (\(\Omega\) any classical family). The property ''isomorphic finite \(\pi\)-subgroups are conjugate'' is isolated for further study, and analogies with universal locally finite groups are explored.