Nonexistence of perfect locally finite minimal non-FC and CC-groups, and a characterization of perfect locally finite barely transitive groups

Abstract In this short note we prove that if G is a perfect locally finite barely transitive group, then the finitary residual of G, namely F (G) =\g G|~| supp (g) | F (G) = { g ∈ G | | supp (g) | ω, is trivial. In particular, we prove that there do not exist any perfect locally finite minimal non- FC and minimal non- CC -groups. This completes the description of minimal non- FC -groups and locally graded minimal non- CC -groups. It is a long-standing problem (see 14, Problem 5. 1 (b) ).