Finite Approximations to Lie Groups

A certain sense in which a finite group may be said to approximate the structure of a metrical group will be discussed. On account of Jordan's theorem on finite groups of linear transformations' it is clear that we cannot hope to approximate a general Lie group with finite subgroups. I shall show that we cannot approximate even with groups which are 'approximately subgroups': in fact the only approximable Lie groups are the compact Abelian groups. The key to the situation is again afforded by Jordan's theorem, but it is not immediately applicable. It is necessary to find representations of the approximating groups whose degree depends only on the group approximated.