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Turkish Journal of Mathematics

ALİ OSMAN ASAR

DOI

10.3906/mat-1508-66

Abstract

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 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-$FC$-groups. Furthermore, some properties of a maximal $p$-subgroup of the finitary symmetric group on $\mathbb{N}^*$ are obtained.

Keywords

Finitary permutation, totally imprimitive, cyclic-block property, homogeneous permutation, $FC$-group

First Page

983

Last Page

997

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