Let G G be a finite group and π \pi a set of primes. We consider the family of subgroups of G : F = { M : M > ⋅ G , [ G : M ] π = 1 , [ G : M ] G:\mathcal {F} = \{ M:M > \cdot G,{[G:M]_\pi } = 1,[G:M] is composite} and denote S π ( G ) = ⋂ { M : M ∈ F } {S_\pi }(G) = \bigcap \left \{ M: M \in \mathcal {F} \right \} if F \mathcal {F} is non-empty, otherwise S π ( G ) = G {S_\pi }(G) = G . The purpose of this note is to prove Theorem. Let G G be a π \pi -solvable group. Then S π ( G ) {S_\pi }(G) has the following properties: (1) S π ( G ) / O π ( G ) {S_\pi }(G)/{O_\pi }(G) is supersolvable. (2) S π ( S π ( G ) ) = S π ( G ) {S_\pi }({S_\pi }(G)) = {S_\pi }(G) . (3) G / O π ( G ) G/{O_\pi }(G) is supersolvable if and only if S π ( G ) = G {S_\pi }(G) = G .

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