For an outer action $alpha$ of a finite group G on a factor M, it was proved that H is a, normal subgroup of G if and only if there exists a finite group F and an outer action $eta$ of F on the crossed product algebra M $ imes$$_{alpha}$ G = (M $ imes$$_{alpha}$ F. We generalize this to infinite group actions. For an outer action $alpha$ of a discrete group, we obtain a Galois correspondence for crossed product algebras related to normal subgroups. When $alpha$ satisfies a certain condition, we also obtain a Galois correspondence for fixed point algebras. Furthermore, for a minimal action $alpha$ of a compact group G and a closed normal subgroup H, we prove $M^{G}$ = ( $M^{H}$)$^{{beta}(G/H)}$for a minimal action $eta$ of G/H on $M^{H}$.f G/H on $M^{H}$.TEX> H/.