I became interested in the following question and realized that it was asked by Niko Bellic in the comment to [previous][1] 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.

  [1]: https://mathoverflow.net/questions/57129/which-finite-groups-have-faithful-complex-irreducible-representations