I became interested in the following question and realized that it was asked by Niko Bellic in the comment to previous question of mine. For which finite groups a complex representation which is free on the complement of the origin does exist? Of course it may be a priori irreducible. In other words, how to describe finite matrix groups for which $A-I$ is invertible for all $A\ne I$ in the group? For Abelian groups only cyclic groups satisfy this property.