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Permutation Groups With Cyclic-Block Property And $Mnfc$-Groups, Ali̇ Osman Asar

Turkish Journal of Mathematics

This work continues the investigation of perfect locally finite minimal non-$FC$-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-$FC$-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-$FC$-subgroup of it is transitive, which are the properties satisfied by a minimal non-$FC$-group. Here a sufficient condition is given for the nonexistence of minimal non-$FC$-groups in this class of permutation groups. In …