Groups generated by two bicyclic units in integral group rings
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Knowledge AreaEconomía Aplicada
SponsorsThe first author has been partially supported by the Onderzoeksraad of Vrije Universiteit Brussel and the Fonds voor Wetenschappelijk Onderzoek (Vlaanderen) and the second by the D.G.I. of Spain and Fundación Séneca of Murcia.We would like to express our gratitude to Victor Jiménez for some helpful conversation on inequality .
PublisherWalter de Gruyter
Bibliographic CitationJESPERS, Eric, RÍO, Ángel, de, RUIZ, Manuel. Groups generated by two bicyclic units in integral group rings. Journal of Group Theory, 5 (4): 493–511, 17 Septiembre 2002. ISSN 1433-5883
Bass cyclic units
Structure of the group
Unidades cíclicas de Bass
Estructuras de grupo
In  Ritter and Sehgal introduced the following units, called the bicylic units, in the unit group U(ZG) of the integral group ring ZG of a finite group G: ¯a;g = 1 + (1 ¡ g)abg; °a;g = 1 + bga(1 ¡ g); where a; g 2 G and bg is the sum of all the elements in the cyclic group hgi. It has been shown that these units generate a large part of the unit group of ZG. Indeed, for most finite groups G, the bicyclic units together with the Bass cyclic units generate a subgroup of finite index in U(ZG) [3, 6]. The Bass cyclic units are only needed to cover a subgroup of finite index in the centre and the group B generated by the bicyclic units contains a subgroup of finite index in a maximal Z-order of each non-commutative simple image Mn(D) of the rational group algebra QG. In particular, if n > 1, then B contains a subgroup of finite index in SLn(O), where O is a maximal order in D; and hence B contains free subgroups of rank two. A next step in determining the structure of U(ZG) is ...
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