A group G is called an FC-group if every conjugacy class of G is finite. FC-groups were first studied by R. Baer and B.H. Neumann and then by many other authors. In this paper we investigate groups G with a finite number of conjugacy classes of infinite size. We denote by D_i the class of groups with i conjugacy classes of infinite size and by D the union of of all the classes D_i, i = 0, 1, 2, . . . . 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 some properties of groups in D are described. Moreover periodic locally graded in D are characterized, they turn out to be either FC-groups or certain finite-by-nilpotent-by-finite groups (a group G is called locally graded if each non-trivial finitely generated subgroup of G contains a proper subgroup of finite index). Finally D_1 groups are studied in detail.
On generalized FC-groups
LONGOBARDI, Patrizia;MAJ, Mercede
2008-01-01
Abstract
A group G is called an FC-group if every conjugacy class of G is finite. FC-groups were first studied by R. Baer and B.H. Neumann and then by many other authors. In this paper we investigate groups G with a finite number of conjugacy classes of infinite size. We denote by D_i the class of groups with i conjugacy classes of infinite size and by D the union of of all the classes D_i, i = 0, 1, 2, . . . . 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 some properties of groups in D are described. Moreover periodic locally graded in D are characterized, they turn out to be either FC-groups or certain finite-by-nilpotent-by-finite groups (a group G is called locally graded if each non-trivial finitely generated subgroup of G contains a proper subgroup of finite index). Finally D_1 groups are studied in detail.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
