NON-SIMPLICITY OF LOCALLY FINITE BARELY TRANSITIVE GROUPS

We answer the following questions negatively: Does there exist a simpl e locally finite barely transitive group (LFBT-group)? More precisely we have: There exists no simple LFBT-group. We also deal with the ques tion, whether there exists a LFBT-group G acting on an infinite set Om ega so that G is a group of finitary permutations on Omega. Along this direction we prove: If there exists a finitary LFBT-group G, then G i s a minimal non-FC p-group. Moreover we prove that: If a stabilizer of a point in a LFBT-group G is abelian, then G is metabelian. Furthermo re G is a p-group for some prime p, G/G' congruent to C-p infinity and G' is an abelian group of finite exponent.