Asar, Ali Osman Permutation groups with cyclic-block property and MNFC-groups. (English) Zbl 1424.20001 Turk. J. Math. 41, No. 4, 983-997 (2017). Summary: 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. Cited in 2 Documents MSC: 20B07 General theory for infinite permutation groups 20B35 Subgroups of symmetric groups 20E25 Local properties of groups 20F24 FC-groups and their generalizations 20B22 Multiply transitive infinite groups Keywords:finitary permutation; totally imprimitive; cyclic-block property; homogeneous permutation; FC-group PDFBibTeX XMLCite \textit{A. O. Asar}, Turk. J. Math. 41, No. 4, 983--997 (2017; Zbl 1424.20001) Full Text: DOI References: [1] Asar AO. On finitary permutation groups. Turk J Math 2006; 30: 101-116. · Zbl 1098.20001 [2] Asar AO. Subgroups of finitary permutation groups. J Group Theory 2008; 11: 229-234. · Zbl 1138.20003 [3] Asar AO. Corrigendum: Subgroups of finitary permutation groups. J Group Theory 2009; 12: 487-489. · Zbl 1167.20301 [4] Asar AO. Totally imprimitive permutation groups with the cyclic-block property. J Group Theory 2011; 14: 127-141. · Zbl 1222.20001 [5] Asar AO. Subgroups of totally imprimitive permutation groups. Comm Algebra 2017; 6: 2690-2707. · Zbl 1388.20004 [6] Belyaev VV, Sesekin, NF. On infinite groups of Miller-Moreno type. Acta Math Acad Sci Hungar 1975; 26: 369-376. · Zbl 0335.20013 [7] Belyaev VV. Minimal non- F C -groups. All Union Symposium Kiev 1980: 97-108. [8] Belyaev VV. On the question of existence of minimal non- F C -groups. Siberian Math J 1998; 39: 1093-1095. · Zbl 0941.20040 [9] Bhattacharjee, M, Machperson, DR, M¨oller, G, Neumann, PM. Notes on Infinite Permutation Groups. Lecture Notes in Mathematics, Vol. 1698. Berlin, Germany: Springer-Verlag, 1998.) [10] Dixon JD, Moretimer, B. Permutation Groups. New York, NY, USA: Springer, 1996. [11] Humphreys JH. A Course in Group Theory. Oxford, UK: Oxford University Press, 1996. · Zbl 0843.20001 [12] Kargapolov MI, Merzljakov, JI. Fundamentals of the Theory of Groups. 2nd ed. Translated from the Russian by RG Burns. New York, NY, USA: Springer, 1979. · Zbl 0549.20001 [13] Kezlan TP, Rhee NH. A characterization of the centralizer of a permutation. Missouri J Math Sci 1999; 11: 158-163. · Zbl 1097.20502 [14] Kuzucuoglu M, Phillips R. Locally finite minimal non- F C -groups. Math Proc Cambridge 1989; 105: 417-420. · Zbl 0686.20034 [15] Leinen F. A reduction theorem for perfect locally finite minimal non- F C -groups. Glasgow Math J 1999; 41: 81-83. · Zbl 0922.20043 [16] Leinen F, Puglisi O. Finitary representations and images of transitive finitary permutation groups. J Algebra 1999; 222: 524-549. · Zbl 0947.20002 [17] Neumann PM. On the structure of standard wreath products of groups. Math Z 1964; 84: 343-373. · Zbl 0122.02901 [18] Neumann PM. The lawlessness of groups of finitary permutations. Arch Math 1975; 26: 561-566. · Zbl 0338.20037 [19] Neumann PM. The structure of finitary permutation groups. Arch Math 1976; 27: 3-17. · Zbl 0324.20037 [20] Pinnock CJE. Irreducible and transitive locally- nilpotent by abelian groups. Arch Math 2000; 74: 168-172. · Zbl 0961.20032 [21] Robinson DJS. A Course in the Theory of Groups. New York, NY, USA: Springer, 1980. [22] Tomkinson MJ. F C -Groups. Boston, MA, USA: Pitman Advanced Publishing Program, 1984. [23] Wiegold J. Groups of finitary permutations. Arch Math 1974; 25: 466-469. · Zbl 0298.20033 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.