Finite groups with generalized Ore supplement conditions for primary subgroups

Abstract

We associate with every group G a set τ(G) of subgroups of G with 1∈τ(G). If H∈τ(G), then we say that H is a τ-subgroup of G. If θ(τ(G))⊆τ(θ(G)) for each epimorphism 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) whenever H≤E≤G and H∈τ(G); regular provided for any group G, whenever H∈τ(G) is a p-group and N is a minimal normal subgroup of G, then |G:NG(H∩N)| is a power of p; Φ-regular (respectively Φ-quasiregular) provided for any primitive group G, whenever H∈τ(G) is a p-group and N is a (respectively abelian) minimal normal subgroup of G, then |G:NG(H∩N)| is a power of p.Let K≤H be subgroups of G and τ a subgroup functor. Then we say that: the pair (K,H) satisfies the F-supplement condition in G if G has a subgroup T such that HT=G and H∩T⊆KZF(T); H is Fτ-supplemented in G if for some τ-subgroup S¯ of G¯ contained in H¯ the pair (S¯,H¯) satisfies the F-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 Fτ-supplemented in G. In particular, we prove the following result.Theorem A.LetFbe a saturated formation containing the classUof all supersoluble groups, E a normal subgroup of G withG/E∈F,X=EorX=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 isUτ-supplemented in G, thenG∈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.