In particular, it is shown that the totally imprimitive permutation p-group satisfying the cyclic-block property that was constructed earlier and its commutator subgroup cannot be minimal non-F C -groups. Furthermore, some properties of a maximal p-subgroup of the finitary symmetric group on N are obtained. | Turk J Math (2017) 41: 983 – 997 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Permutation groups with cyclic-block property and M N F C -groups Ali Osman ASAR∗ Yargı¸c Sokak 11/6 Cebeci, Ankara, Turkey Received: • Accepted/Published Online: • Final Version: Abstract: This work continues the investigation of perfect locally finite minimal non- F C -groups in totally imprimitive permutation p -groups. At present, the class of totally imprimitive permutation p -groups satisfying the cyclic-block property is known to be the only class of p -groups having common properties with a hypothetical minimal non- F C group, because a totally imprimitive permutation p -group satisfying the cyclic-block property cannot be generated by a subset of finite exponent and every non- F C -subgroup of it is transitive, which are the properties satisfied by a minimal non- F C -group. Here a sufficient condition is given for the nonexistence of minimal non- F C -groups in this class of permutation groups. In particular, it is shown that the totally imprimitive permutation p -group satisfying the cyclic-block property that was constructed earlier and its commutator subgroup cannot be minimal non- F C -groups. Furthermore, some properties of a maximal p -subgroup of the finitary symmetric group on N∗ are obtained. Key words: Finitary permutation, totally imprimitive, cyclic-block property, homogeneous permutation, F C -group 1. Introduction Let Ω be a nonempty (infinite) set. A permutation g on Ω is called finitary if its support supp(g) is finite. The set of all the finitary permutations on Ω forms a normal subgroup of the symmetric group Sym(Ω) and is called the restricted symmetric group on Ω. It is denoted by F Sym(Ω) . A subgroup of F Sym(Ω) is called a finitary permutation group on Ω . Let G be a transitive finitary permutation group on Ω , where Ω is infinite. If G
