THE EXISTENCE OF LARGE OMEGA(1)-HOMOGENEOUS BUT NOT OMEGA-HOMOGENEOUSPERMUTATION-GROUPS IS CONSISTENT WITH ZFC+GCH

Denote by Perm (lambda) the group of all permutations of a cardinal la mbda. A subgroup G of Perm (lambda) is called kappa-homogeneous if and only if for all X, Y epsilon[lambda](k)appa there is a g epsilon G wi th g'' X = Y. We show that if either (i) lozenge(+) holds and we add o mega(1) Cohen reals to the ground model, or (ii) we add 2(o)mega(1) Co hen reals to the ground model, then in the generic extension for each lambda greater than or equal to omega(2) there is an omega(1)-homogene ous subgroup of Perm(lambda) which is not omega-homogeneous.