THE G-COMPACTIFICATIONS OF A LOCALLY COMPACT HAUSDORFF SPACE

Let G(X) denote the group of autohomeomorphisms of the locally compact Tychonov space X, and let C(X) denote the lattice of its compactifications. We demonstrate that information about G(X) and C(X) can be sensibly combined to yield a complete sublattice C(G)(X), the elements of which are those compactifications of X which 'preserve' the action of G(X) on X. We derive a condition which is sufficient to ensure that 'G(X) = G(Y)' is equivalent to 'X = Y', whenever X and Y are also compact. Finally, we demonstrate that C(X) = C(G)(X) precisely when beta-X X is rigid.