Abstract
Let (sic) be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, (sic) contains exactly one and only one Sylow p-subgroup of G, say G(p). Let C be a nonempty subset of G. A subgroup H of G is said to be C-(sic)-permutable (conjugate-(sic)-permutable) subgroup of G if there exists some x is an element of C such that H(x)G(p) = G(p)H(x), for all G(p) is an element of (sic). We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-(sic)-permutable (conjugate-(sic)-permutable) subgroups of G. Our results improve and generalize several results in the literature.