Groups with soluble minimax conjugate classes of subgroups

Document Type : Research Article


Mathematics Department, University of Wuerzburg, Wuerzburg, Germany


A classical result of Neumann characterizes the groups in which each subgroup has finitely many conjugates only as central-by-finite groups. If X is a class of groups, a group G is said to have X-conjugate classes of subgroups if G/coreG(NG(H)) 2 X for each subgroup H of G. Here we study groups which have soluble minimax conjugate classes of subgroups, giving a description in terms of G/Z(G). We also characterize FC-groups which have soluble minimax conjugate classes of subgroups.


[1] Baer, R., Finiteness properties of groups, Duke Math. J. 15(1948), 1021-1032.
[2] Franciosi, S., Giovanni F. de and Tomkinson, M.J., Groups with polycyclic-by-finite conjugacy classes, Boll. Unione Mat. Ital. 4B(1990), 35-55.
[3] Franciosi, S., Giovanni F. de and Kurdachenko, L., Groups whose proper quotients are F C-groups, J. Algebra 186(1995), 544-577.
[4] Kurdachenko, L., On groups with minimax conjugacy classes, Infinite groups and adjoining algebraic structures, Naukova Dumka, Kiev, 1993, 160-177.
[5] Kurdachenko L. and Otal, J., Frattini properties of groups with mini-max conjugacy classes, Topics in Infinite Groups, Quad. di Mat. Vol.8 Caserta(2000), 223-235.
[6] Kurdachenko, L., Otal J. and Soules, P., Polycyclic-by-finite conjugate classes of subgroups, Comm. Algebra 32(2004), 4769-4784.
[7] Kurdachenko, L. and Otal, J., Groups with Chernikov classes of conjugate subgroups, J. Group Theory 8(2005), 93-108.
[8] Neumann, B.H., Groups with finite classes of conjugate subgroups, Math. Z. 63(1955), 76-96.
[9] Polovicky, Ya.D., Groups with extremal classes of conjugate elements, Sibirsk. Mat. Z. 5(1964), 891-895.
[10] Polovicky, Ya.D., The periodic groups with extremal classes of conjugate abelian subgroups, Izvestija VUZ, ser Math. 4(1977), 95-101.
[11] Robinson, D.J.S., Finiteness Conditions and Generalized Soluble Groups. Berlin, Springer-Verlag, 1972.