  If V is a representation of a linear algebraic group G, a set S of
G-invariant regular functions on V is called separating if the following holds:
If two elements v,v' from V can be separated by an invariant function, then
there is an f from S such that f(v) is different from f(v'). It is known that
there always exist finite separating sets. Moreover, if the group G is finite,
then the invariant functions of degree <= |G| form a separating set. We show
that for a non-finite linear algebraic group G such an upper bound for the
degrees of a separating set does not exist. If G is finite, we define b(G) to
be the minimal number d such that for every G-module V there is a separating
set of degree less or equal to d. We show that for a subgroup H of G we have
b(H) <= b(G) <= [G:H] b(H)$, and that b(G) <= b(G/H) b(H)$ in case H is normal.
Moreover, we calculate b(G) for some specific finite groups.
