On some integral representations of groups and global irreducibility

Arithmetic aspects of integral representations of finite groups and their irreducibility are considered with a focus on globally irreducible representations and their generalizations to arithmetic rings. Certain problems concerning integral irreducible two-dimensional representations over number rings are discussed. Let K be a finite extension of the rational number field and OK the ring of integers of K. Let G be a finite subgroup of GL(2, K), the group of (2 × 2)-matrices over K. We obtain some conditions on K for G to be conjugate to a subgroup of GL(2, O_K).