Finite subgroups of units of group algebras. I (Q797665)

The following question is considered: If G is a finite group and K an algebraic number field (with ring of integers R), what can be said about the finite subgroups of the unit group of KG which contain G? The main result is proved in section 2, using the existence of an involution for the coefficient ring. Let K have a complex involution *. Define an involution J of KG by \(J(\sum a_ gg)=\sum a^*_ gg,\) and, for a subring A of KG, let \(U_ J(A)=\{x\in A| \quad xJ(x)=1\}.\) Theorem 2. If A is an R-order in KG containing RG, then \(U_ J(A)\) is a finite group containing G. As A ranges over all maximal R-orders containing RG, \(U_ J(A)\) ranges over all maximal finite subgroups of KG containing G. Some interesing consequences are given and the case in which G is a dihedral group is studied in some detail in section 3. This method for the study of finite subgroups of KG arises from a nice result in algebraic number theory.