Finite groups with generalized Ore supplement conditions for primary subgroups

We associate with every group G   a set τ(G)τ(G) of subgroups of G   with 1∈τ(G)1∈τ(G). If H∈τ(G)H∈τ(G), then we say that H is a τ-subgroup of G  . If θ(τ(G))⊆τ(θ(G))θ(τ(G))⊆τ(θ(G)) for each epimorphism θ:G→G⁎θ:G→G⁎, then we say that τ is a subgroup functor. We say also that a subgroup functor τ is: hereditary   provided H∈τ(E)H∈τ(E) whenever H≤E≤GH≤E≤G and H∈τ(G)H∈τ(G); regular provided for any group G  , whenever H∈τ(G)H∈τ(G) is a p-group and N is a minimal normal subgroup of G  , then |G:NG(H∩N)||G:NG(H∩N)| is a power of p; Φ-regular (respectively Φ-quasiregular) provided for any primitive group G  , whenever H∈τ(G)H∈τ(G) is a p-group and N is a (respectively abelian) minimal normal subgroup of G  , then |G:NG(H∩N)||G:NG(H∩N)| is a power of p.Let K≤HK≤H be subgroups of G and τ   a subgroup functor. Then we say that: the pair (K,H)(K,H) satisfies the FF-supplement condition in G if G has a subgroup T   such that HT=GHT=G and H∩T⊆KZF(T)H∩T⊆KZF(T); H   is FτFτ-supplemented in G if for some τ  -subgroup S¯ of G¯ contained in H¯ the pair (S¯,H¯) satisfies the FF-supplement condition in G¯, where G¯=G/HG and H¯=H/HG.In this paper we study the structure of a group G under the condition that some primary subgroups of G   are FτFτ-supplemented in G. In particular, we prove the following result.Theorem A.Let  FFbe a saturated formation containing the class  UUof all supersoluble groups, E a normal subgroup of G with  G/E∈FG/E∈F,  X=EX=Eor  X=F⁎(E)X=F⁎(E), and τ a regular or hereditary Φ-regular subgroup functor. Suppose that every τ-subgroup of G contained in X is subnormally embedded in G. If every maximal subgroup of every non-cyclic Sylow subgroup of X is  UτUτ-supplemented in G, then  G∈FG∈F. Moreover, in the case when τ is regular, then every chief factor of G below E is cyclic.The results in this paper not only cover and unify a long list of some known results but also cause a wide series of new results.