J. Korean Math. Soc. 47 (2010), No. 6, pp. 1239–1252 DOI 10.4134/JKMS.2010.47.6.1239 CHARACTERIZATION OF CENTRAL UNITS OF ZA n Tevfik Bilgin, Necat Gorentas, and I. Gokhan Kelebek Abstract. The structure of V (Z(ZAn)) is known when n ≤ 6. If n =5 or 6, then a complete set of generators of V (Z(ZAn)) has been deter- mined. In this study, it was shown that V (Z(ZAn)) is trivial when n =7, 8 or 9 and it is generated by a single unit u when n = 10 or 11. This unit u is characterized explicitly for n = 10 or 11 by using irreducible characters of An. 1. Introduction Let V = V (ZG) denote the group of normalized units of the integral group ring ZG of a finite group G. Let Z (V ) denote the subgroup of the central units of V and N V (G) denote the normalizer of G in the normalized units V (ZG). In [4], the following is considered as the normalizer property: N V (G)= GZ (V )? Verification of the normalizer property reduces the problem to the construc- tion of central units of normalized units of ZG [4, 5]. On the other hand, in order to find a counter example one must find a unit u ∈N V (G) \Z (V ). In both cases, construction of Z (V ) is important. Let V (Z (ZG)) denote the normalized units of the center of ZG. Both of them are defined as follows [2]: Z (V )= Z (V (ZG)) = V (ZG) ∩Z (ZG)= V (Z (ZG)). The problem of finding the full structure of V (Z (ZG)), including a complete set of generators, has been determined for only a small number of special cases. When G is finite, Patay [8] proved the following theorem satisfying necessary and sufficient conditions for U (Z (ZG)) to be trivial. Proposition 1.1. Let G be a finite group. All central units of ZG are trivial if and only if for every x ∈ G and for every j ∈ N relatively prime to |G| ,x j is conjugate to x or x -1 . Ritter and Sehgal constructed a finite set of generators for a subgroup of finite index in U (Z (ZG)) for a finite group G [9]. On the other hand, Jespers, Received February 26, 2009; Revised August 21, 2009. 2000 Mathematics Subject Classification. 16S34, 16U60. Key words and phrases. normalizer, centralizer, generators of central units. c 2010 The Korean Mathematical Society 1239