# On some integral representations of finite groups

*by Dr Dmitry Malinin *

*Institution:*University of the South Pacific

*Date: Tue 9th January 2007*

*Time: 3:15 PM*

*Location: Room 213, Richard Berry Building, The University of Melbourne*

*Abstract*: We consider the natural action of Galois groups on finite arithmetic groups and finite subgroups of GL(n,R) for orders R in number fields.

Let K/Q be a finite Galois extension with the ring of integers O_K and Galois group \Gamma. We consider finite \Gamma-stable subgroups G< GL_n(O_K) and prove that they are generated by matrices with coefficients in O_{K_ab}, K_ab the maximal abelian subextension of K over Q. This implies in particular a positive answer to a conjecture of J. Tate on the classification of p-divisible groups over Z and answers also a longstanding question of Y. Kitaoka on totally real scalar extensions of positive definite integral quadratic lattices.

*For More Information:* Dr. Lawrence Reeves email: L.Reeves@ms.unimelb.edu.au