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
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).