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

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|~| supp (g)|<ω \ 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)]).