**Abstract:**
Let $k$ be a field, L_n = k[x_1^{\pm 1}, \dots, x_n^{\pm 1}]
be the Laurent polynomial ring in n variables and
G be a group of k-algebra automorphisms of L_n. We give
a necessary and sufficient condition for the ring of invariants
L_n^G to have a SAGBI basis. We show that if this condition is satisfied
then L_n^G has a SAGBI basis relative to any choice of
coordinates in L_n and any term order.