An element x of a group G is called an FC-element if the conjugacy class of x is finite. A group is called an FC-group if all its elements are FC-elements. FC-groups were first studied by R. Baer and B.H. Neumann and then by many other authors. We denote by D_i the class of groups with i conjugacy classes of infinite size. The D_0-groups are exactly the FC-groups and the infinite dihedral group belongs to D_2. Constructing HNN-extension, G. Higman, B.H. Neumann and H. Neumann have shown that any torsion-free group can be embedded in a group G, in which all non-trivial elements are conjugate. Since each such group G belongs to D_1, the structure of D_1-groups can be very complicated. In this paper we study groups in D_2. Our aim is to determine the structure of such groups G under the assumption that G/F is periodic, where F = FC(G) denotes the FC-center of G i.e. the set of all FC-elements of G.
On groups with two infinite conjugacy classes
LONGOBARDI, Patrizia;MAJ, Mercede
2007
Abstract
An element x of a group G is called an FC-element if the conjugacy class of x is finite. A group is called an FC-group if all its elements are FC-elements. FC-groups were first studied by R. Baer and B.H. Neumann and then by many other authors. We denote by D_i the class of groups with i conjugacy classes of infinite size. The D_0-groups are exactly the FC-groups and the infinite dihedral group belongs to D_2. Constructing HNN-extension, G. Higman, B.H. Neumann and H. Neumann have shown that any torsion-free group can be embedded in a group G, in which all non-trivial elements are conjugate. Since each such group G belongs to D_1, the structure of D_1-groups can be very complicated. In this paper we study groups in D_2. Our aim is to determine the structure of such groups G under the assumption that G/F is periodic, where F = FC(G) denotes the FC-center of G i.e. the set of all FC-elements of G.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
