THE HOMOMORPHIC IMAGES OF INFINITE SYMMETRICAL GROUPS

We give a solution of De Bruijn's problem as to which homomorphic imag es of an infinite symmetric group S(kappa) are embeddable into S(kappa ). We prove (in ZF + V = L) that S(kappa)/S(lambda)(kappa) can be embe dded into S(kappa) if and only if lambda < Cf(kappa). We also discuss F. Clare's problem whether S(kappa)/S(kappa)(kappa) is a universal gro up. We prove that it is consistent with ZFC that S(omega)/S(omega)(ome ga)) is not universal. However S(kappa)/S(kappa)(kappa) is almost univ ersal, since the group S(kappa+)(kappa+) of all bounded permutations O f kappa+ embeds into it when kappa is regular.