F-Perfect Groups With Some FC-Subgroups

Arıkan A.

International Conference on Mathematics and Mathematics Education (ICMME-2017), Harran University, Şanlıurfa, 11-13 May 2017, Şanlıurfa, Türkiye, 11 Mart - 13 Aralık 2017, ss.73-74

  • Basıldığı Şehir: Şanlıurfa
  • Basıldığı Ülke: Türkiye
  • Sayfa Sayıları: ss.73-74


 We deal with the groups having FC-subgroups and define the class of groups. We provide two points of view and consider such a group as a subgroup of for some prime p, the McLain groups, or represent as a finitary permutation group on an infinite set. Having certain FC-subgroups in groups may provide some opportunities to figure out the structure of the groups (see [1,Theorem 1.1(b)] for example). In this vork we give certain useful results. The following is a slightly generalized form of [4,Theorem 2.4] (see 4 for the definition of the notion “locally degree preserving"). In this study, we obtained the following results: Let be a perfect locally finite p-group for some prime p. If there exits such that is an -group for every then there exists a locally degree-preserving embedding of an epimorphic image of into for some prime p. The above result provides a restriction our attention to McLain goups for some perfect groups (of course if such groups exist) Define the class of groups as follows: The normal closure of every two generated subgroup of the group is an FCgroup. Let be an infinitely generated locally finite -perfect-p-group for some prime p with trivial center. If is a -group then has an epimorphic image which can be represented as a finitary permutation group. Key Words: FC-subgroups, -perfect-p-group, McLain group, finitary permutation group, perfect group, -group. International Conference on Mathematics and Mathematics Education (ICMME-2017), Harran University, Şanlıurfa, 11-13 May 2017 74 REFERENCES [1] A.O. Asar, On infinitely generated groups whose proper subgroups are solvable. J. of Algebra. 399 (2014), 870-886. [2] V.V. Belyaev, On the question of existence of minimal non-FC-groups. Siberian Mathematical Journal; 39 No. 6, 1093-1095.; translated from [3] J.C. Lennox, D.J.S. Robinson, The theory of infinite soluble groups. Clarendon Press, Oxford, 2004. [4] F. Leinen, O. Puglisi, Unipotent finitary linear groups. J. London Math. Soc. 48 (1993), 59-76. [5] U. Meierfrankenfeld, R.E. Phillips, O. Puglisi. Locally solvable finitary linear groups. J. London Math. Soc. 47 (1993), 31-40. [6] D.J.S. Robinson, Finiteness conditions and generalized soluble groups . Vol 2.SpringerVerlag, Berlin--Heidelberg, (1972). [7] M. J. Tomkinson, FC-groups. Pitman Advanced Pub. Program, London--Boston--Melbourne, (1984).