Throughout this note we will denote by p a fixed prime number. All groups considered will be finite.

In the theory of groups it is well known that the formula “subnormal + pronormal = normal”. In this note, we define an embedding property of subgroups such that the previous formula with p-subnormal instead subnormal is also true. We call this property, which is stronger than pronormality, p-pronormality.

We give tests for p-pronormality that will be used in inductive proofs. By means of these we can show that the introduced concept is essentially new, in the sense that p-pronormahty is not a particular case of the already known property of ℑpronormality for any saturated formation ℑ.

Recall that if G is a group, P a Sylow subgroup of G and H Ⅴ G it is said that P reduces into H if P ∩ H is a Sylow subgroup of H.

Definition 1 Let G be a group and H a subgroup of G. Then H is said to be p-pronormal in G if each Sylow p-subgroup P of G reduces into an unique conjugate subgroup of H in G; i.e. if P ∩ H ∈ SylP(H) and P ∩ Hg ∈ Sylp(Hg), then g ∈ NG(H).