Consideration of subgroups of the group of all subsets of N={1, 2, ..., n} with symmetric difference as the operation leads to new inequalities on correlations for a generalized Ising ferromagnet. An upper bound for the rate of change of 〈σR〉 with respect to JS in terms of correlations and a new, brief proof for the monotonicity of 〈σR〉 as a function of JS are given. © 1969 Springer-Verlag.