Near-representations of finite groups L´aszl´oBabai ∗ Katalin Friedl † Andr´asLuk´acs ‡ June 19, 2003 Abstract A near-representation of a group G is a map M : G → GL(V ), where V is a complex linear space and M gh ≈ M g M h for all g,h ∈ G, i.e. M is approximately a homomorphism. Analogously to the repre- sentation theory of finite groups, we investigate the properties of near- representations and show that they are equivalent to unitary near- representations, and can be split to stably irreducible constituents. This will imply that near-representations are necessarily representa- tions with a small error. The required error bound of the multipli- cation does not depend on the order of the group G, only on the dimension and size of the near-representation. 1 Introduction Let GL(V) be the space of invertible linear operators of finite dimensional vector space V over the complex numbers C. A map L of a finite group G associates a linear operator L g ∈ GL(V) to each group element g ∈ G is called a representation of the group G if the map L is a homomorphism, i.e. L gh = L g L h for all g,h ∈ G. * University of Chicago and E¨ otv¨ os University. e-mail: laci@cs.uchicago.edu † Computer and Automation Research Institute Hungarian Academy of Sciences. Sup- ported in part by OTKA. e-mail: friedl@sztaki.hu ‡ Computer and Automation Research Institute Hungarian Academy of Sciences. Sup- ported in part by OTKA Txxxx and D34560. e-mail: alukacs@sztaki.hu 1