Groups whose proper subgroups of infinite rank have polycyclic-by-finite conjugacy classes

Document Type: Research Paper

Authors

University Setif 1

Abstract

A group $G$ is said to be a $(PF)C$-group or to have polycyclic-by-finite conjugacy classes, if $G/C_{G}(x^{G})$ is a polycyclic-by-finite group for all $x\in G$. This is a generalization of the familiar property of being an $FC$-group. De Falco et al. (respectively, de Giovanni and Trombetti) studied groups whose proper subgroups of infinite rank have finite (respectively, polycyclic) conjugacy classes. Here we consider groups whose proper subgroups of infinite rank are $(PF)C$-groups and we prove that if $G$ is a group of infinite rank having a non-trivial finite or abelian factor group and if all proper subgroups of $G$ of infinite rank are $(PF)C$-groups, then so is $G$. We prove also that if $G$ is a locally soluble-by-finite group of infinite rank which has no simple homomorphic images of infinite rank and whose proper subgroups of infinite rank are $(PF)C$-groups, then so are all proper subgroups of $G$.

Keywords

Main Subjects

References

[1] O. D. Artemovych, Minimal non-$PC$-groups, Algebra Discrete Math., 18 (2014) 1-7.

[2] V. V. Beljaev and N. F. Sesekin, In nite groups of Miller-Moreno type, Acta Math. Acad. Sci. Hungar., 26 (1975) 369-376.

[3] V. V. Beljaev, Minimal non-$FC$-groups, Sixth All-Union Symposium on Group Theory ( Cerkassy, 1978) (Russian), 221, "Naukova Dumka", Kiev, 1980 97-102.

[4] M. Bouchelaghem and N. Trab elsi, On minimal non- MrC -groups, Ric. Mat., 62 (2013) 97-105.

[5] N. S. Chernikov, A theorem on groups of nite special rank, Ukrainian Math. J., 42 (1990) 855-861.

[6] M. R. Dixon, M. J. Evans and H. Smith, Goups with all prop er subgroups nilpotent-by- nite rank, Arch. Math., 75 (2000) 81-91.

[7] M. De Falco, F. de Giovanni, C. Musella and N. Trabelsi, Groups whose proper subgroups of in nite rank have nite conjugacy classes, Bul l. Aust. Math. Soc., 89 (2014) 41-48.

[8] F. de Giovanni, In nite groups with rank restrictions on subgroups, Problems in the theory of representations of algebras and groups, 3139, Part 25, Zap. Nauchn. Sem. POMI, 414, POMI, St. Petersburg (2013).

[9] F. de Giovanni and M. Tromb etti, Groups whose prop er subgroups of in nite rank have polycyclic conjugacy classes, (to app ear).

[10] S. Franciosi, F. de Giovanni and M. J. Tomkinson, Groups with polycyclic-by- nite conjugacy classes, Boll. Un. Mat. Ital. B (7), 4 (1990) 35-55.

[11] L. A. Kurdachenko, On groups with minimax conjugacy classes, In: In nite groups and adjoining algebraic struc- tures, (Naukova Dumka, Kiev), 1999 160-177.

[12] L. A. Kurdachenko and J. Otal, Frattini properties of groups with minimax conjugacy classes, Topicsin In nite Groups, Topics in in nite groups, Quad. Mat., 8, Dept. Math., Seconda Univ. Napoli, Caserta, 2001 221-237.

[13] J. Otal and J. M. Pe ~na, Minimal non- CC -groups, Comm. Algebra, 16 (1988) 1231-1242.

[14] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Springer Verlag, New York-Berlin, 1972.