Session 14. Group Rings and Related Topics

On infinite groups with finitely many conjugacy classes

Jan Krempa, University of Warsaw, Poland
We consider here infinite groups with a finite number of conjugacy classes (FNCC-groups). Among results on these groups we give a criterion for a wreath product of FNCC-groups to be an FNCC-group. Our motivation goes from some results on endomorphisms of \(C[G]\) and finite generation of the group \(U(C[G]),\) where \(C\) is a commutative, but not reduced ring, and \(G\) is a u.p.-group. The case of \(End_C(C[G])\) and \(U(C[G])\) when \(C\) is reduced is well understood.
References
  1. Yu.N. Gorchinskii, Groups with a finie number of conjugacy classes, Mat. Sbornik N.S. 31 (1952), 167-182.
  2. G. Higman, B.H. Neumann, H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247-254.
  3. J. Krempa, Homomorphisms of Group Rings, in: Banach Center Publications, vol. 9, PWN, Warszawa 1982, 233-255.
  4. J. Krempa, On finite generation of unit groups for group rings, in: Groups '93 Galway/St. Andrews, Vol. 2, London Math. Soc. Lecture Note Ser 212, Cambridge Univ. Press, Cambridge, 1995, 352-367.
  5. J. Krempa, O. MacedoĊ„ska, W. Tomaszewski, On finite number of conjugacy classes in groups, Comm. Algebra 42(12) (2014), 5170-5179.
  6. A. Strojnowski, A note on u.p. groups, Comm. Algebra 8(3) 1980, 231-234.
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