Unitary units in modular group algebras
Abstract
Let p be a prime, K a field of characteristic p, G a locally finite pgroup, KG the group algebra, and V the group of the units of KG with augmentation 1. The antiautomorphism g\mapsto g^{1} of G extends linearly to KG; this extension leaves V setwise invariant, and its restriction to V followed by v\mapsto v^{1} lives an automorphism of V. The elements of V fixed by this automorphism are called unitary; they form a subgroup. Our first theorem describes the K and G for which this subgroup is normal in V. For each element g in G, let \bar{g} denote the sum (in KG) of the distinct powers of g. The elements 1+(g1)h\bar{g} with g,h\in G are the bicyclic units of KG. Our second theorem describes the K and G for which all bicyclic units are unitary.
 Publication:

arXiv eprints
 Pub Date:
 November 2007
 arXiv:
 arXiv:0711.0097
 Bibcode:
 2007arXiv0711.0097B
 Keywords:

 Mathematics  Rings and Algebras;
 Mathematics  Group Theory;
 16U60;
 16S34;
 20C07
 EPrint:
 12 pages