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A Schmidt group is a non-nilpotent group whose every proper subgroup is nilpotent. A subgroup A of a group G is called OS-propermutablein G if there is a subgroup B such that G = N G (A) B , where AB is a subgroup of G and A permutes with all Schmidt subgroups of B. We proved p -solubility of a group in which a Sylow p -subgroup is OS-propermutable, where p ≥ 7 7. For p < 7 all non-Abelian composition factors of such group are listed. [ABSTRACT FROM AUTHOR]

Let G be a finite group and G be a split extension of A by B, that is, G is a semidirect product: G = A ⋊ B , where A and B are subgroups of G. Under the condition that B permutes with every maximal subgroup of Sylow subgroups of A, every maximal subgroup of A or every nontrivial normal subgroup of A, we prove that the supersolvable residual of G is the product of the supersolvable residuals of A and B. [ABSTRACT FROM AUTHOR]

We say a group G = AB is the totally semipermutable product of subgroups A and B if every Sylow subgroup P of A is totally permutable with every Sylow subgroup Q of B whenever gcd (| P | , | Q |) = 1 . Products of pairwise totally semipermutable subgroups are studied in this article. Let U denote the class of supersoluble groups and D denote the formation of all groups which have an ordered Sylow tower of supersoluble type. We obtain the F -residual of the product from the F -residuals of the pairwise totally semipermutable subgroups when F is a subgroup-closed saturated formation such that U ⊆ F ⊆ D . [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called c-normal if there exists a normal subgroup N in G such that G = HN and H ∩ N ≤ c o r e G (H) , the largest normal subgroup of G contained in H. c-Normality is a weaker form of normality, introduced by Y.M. Wang, that has led to interesting results and structural criteria of finite groups. In this paper we study c-normality in the coprime action setting so as to obtain several solvability and p-nilpotency criteria in terms of certain subsets of maximal invariant subgroups of a group or of its Sylow subgroups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H a subgroup of G. We say that H is pronormal in G if for every g ∈ G , the subgroups H and H g are conjugate in ⟨ H , H g ⟩ ; H is called weakly pronormal in G if there exists a subgroup K of G such that G = HK and H ∩ K is pronormal in G. In this paper, we investigate the structure of G under the assumption that certain subgroups of G are weakly pronormal in G. Our results improve and generalize new recent results in the literature. [ABSTRACT FROM AUTHOR]

Let p be a prime number. A positive integer m is said to be a p ′ -number provided p ∤ m . Let G be a finite group, and let H be a subgroup of G. We say that H is weakly s p -permutable in G if G has a subnormal subgroup K such that G = HK, H sG ≤ K and | H ∩ K : H sG | is a p ′ -number, where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We study the structure of a finite group G of even order all of whose fourth maximal subgroups are weakly s 2 -permutable in G. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be an I C Φ -subgroup of G if H ∩ [ H , G ] ≤ Φ (H) . We prove the p-nilpotency of a finite group G under the assumption that certain p-subgroups of G are I C Φ -subgroups of G. Our main result generalizes and extends some recent work of Gao and Li [1]. [ABSTRACT FROM AUTHOR]

Let σ = { σ i | i ∈ I } be some partition of the set of all primes P and G be a finite group. A group is said to be σ -primary if it is a finite σ i -group for some i. A subgroup A of G is said to be σ -subnormal in G if there is a subgroup chain A = A 0 ≤ A 1 ≤ ⋯ ≤ A t = G such that either A i - 1 ⊴ A i or A i / (A i - 1) A i is σ -primary for all i = 1 , ... , t . A subgroup S of G is m- σ -permutable in G if S = ⟨ M , B ⟩ for some modular subgroup M and σ -permutable subgroup B of G. We say that a subgroup H of G is m- σ -embedded in G if there exist an m- σ -permutable subgroup S and a σ -subnormal subgroup T of G such that H G = H T and H ∩ T ≤ S ≤ H , where H G = ⟨ H x | x ∈ G ⟩ is the normal closure of H in G. In this paper, we study the properties of m- σ -embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is called an I C Φ -subgroup of G if H ∩ [ H , G ] ≤ Φ (H) . We investigate the structure of a finite group G under the assumption that some subgroups of G are I C Φ -subgroups of G, and some results of p-nilpotency and supersolvability of finite groups are obtained. [ABSTRACT FROM AUTHOR]

A subgroup A of a finite group G is said to be NS-supplemented in G, if there exists a subgroup B of G such that G=AB and whenever X is a normal subgroup of A and p ∈ π (B) , there exists a Sylow p-subgroup Bp of B such that X B p = B p X . In this paper, we prove the supersolubility of a group G in the following cases: every non-cyclic Sylow subgroup of G is NS-supplemented in G; G is soluble and all maximal subgroups of every non-cylic Sylow subgroup of G are NS-supplemented in G. The solubility of a group with NS-supplemented maximal subgroups is obtained. [ABSTRACT FROM AUTHOR]

Throughout this paper, all groups are finite. σ = { σ i ∣ i ∈ I } is some partition of the set of all primes P , and σ (n) = { σ i ∣ σ i ∩ π (n) ≠ ∅ } for any n ∈ N . The natural numbers n and m are called σ -coprime if σ (n) ∩ σ (m) = ∅ . Let t > 1 be a natural number and let F be a class of groups. Then we say that F is Γ t σ -closed (respectively weakly Γ t σ -closed) provided F contains each finite group G which satisfies the following conditions: G has subgroups A 1 , ... , A t ∈ F such that G = AiAj for all i ≠ j ; The indices | G : N G (A 1) | , ... , | G : N G (A t) | (respectively the indices | G : A 1 | , ... , | G : A t - 1 | , | G : N G (A t) | ) are pairwise σ -coprime. We study properties and some applications of (weakly) Γ t σ -closed classes of finite groups. [ABSTRACT FROM AUTHOR]

We prove a p-nilpotency criterion for finite groups in terms of the element orders of its p′-reduced sections that extends a nilpotency criterion by Tărnăuceanu. [ABSTRACT FROM AUTHOR]

Let σ={σi|i∈I} be some partition of the set of all primes P, G be a finite group and σ(G)={σi|σi∩π(G)≠∅}. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σi-subgroup of G and H contains exactly one Hall σi-subgroup of G for every σi∈σ(G). A subgroup H of G is σ-permutable in G if G possesses a complete Hall σ-set H such that HAx= AxH for all A∈H and all x∈G. We say that a subgroup H of G is n-σ-embedded in G if there exists a normal subgroup T of G such that HT is σ-permutable in G and H∩T≤HσG, where HσG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G. In this paper, we study the properties of the n-σ-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized. [ABSTRACT FROM AUTHOR]

Let σ={πi∣i∈I and πi∩πj=∅ for all i≠j} be a partition of the set of all primes into mutually disjoint subsets. We consider subgroups that permute with given sets of πi -maximal subgroups for all πi∈σ. In particular, we show that such subgroups form a sublattice of the lattice of all subgroups of a finite group. As corollaries we obtain some well known results about S-permutable subgroups. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is called an $${\mathcal{H}}$$ -subgroup in G if $${N_{G}(H)\cap H^{g} \leq H}$$ for all $${g\in G}$$ . A subgroup H of G is called weakly c-supplemented in G if G has a subgroup K such that G = HK and $${ H\cap K}$$ is an $${\mathcal{H}}$$ -subgroup in G. We investigate the structure of finite groups by means of weakly c-supplemented subgroups. Some recent results about supersolvability of finite groups are generalized to a saturated formation containing the class of all supersolvable groups. [ABSTRACT FROM AUTHOR]

Let $${ \mathcal {F}}$$ be a saturated formation and G a finite group such that $${N_{G} (H^{\mathcal {F}})/C_{G} (H^{\mathcal {F}})\cong Inn(H^{\mathcal {F}})}$$ for every subgroup H of G. If the minimal non- $${ \mathcal {F}}$$ -group is soluble, then $${G \in \mathcal {F}}$$ . [ABSTRACT FROM AUTHOR]

A $${\mathfrak{U}_m}$$ -subnormal subgroup of a finite group is introduced, some new characterizations of the hypercyclically embedded subgroups of a finite group are obtained and a series of known results is generalized. [ABSTRACT FROM AUTHOR]

The finite p-groups such that, whenever H < G and $${|H|^{2}p < |G|}$$ , then $${H \triangleleft G}$$ , are classified. Nonnilpotent groups G such that, whenever H < G and $${|H|^{2}p < |G|}$$ , where p is a minimal prime divisor of $${|G|}$$ , then $${H \triangleleft G}$$ , are also classified. In this way we give an answer to Problems 1 and 7 posed in a paper of Y. Berkovich. By using generalizations of these problems we also obtain some new criteria for a group to be a solvable T-group. [ABSTRACT FROM AUTHOR]

A complete classification of finite biprimary groups in which every 3-maximal subgroup is $${\mathfrak{U}}$$ -subnormal is given. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H a subgroup of G. We say that H is s-permutable in G if HP = PH for all Sylow subgroups P of G; H is s-semipermutable in G if HP = PH for all Sylow subgroups P of G with (| P|, | H|) = 1. Let H G be the subgroup of H generated by all those subgroups of G which are s-permutable in G and H the intersection of all such s-permutable subgroups of G contain H. We say that H is nearly s-embedded in G if G has an s-permutable subgroup T such that H = HT and $${H \cap T \leqq H_{ssG}}$$ , where H is an s-semipermutable subgroup of G contained in H. In this paper, we study the structure of a finite group G under the assumption that some subgroups of prime power order are nearly s-embedded in G. A series of known results are improved and extended. [ABSTRACT FROM AUTHOR]

In this paper, we find and correct some errors in the proof of the theorem in [], and give a new proof of the main results. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is said to be SS-permutable in G if H has a supplement K in G such that H permutes with every Sylow subgroup of K. A finite group G is called an SST-group if SS-permutability is a transitive relation on the set of all subgroups of G. The structure of SST-groups is investigated in this paper. [ABSTRACT FROM AUTHOR]

Suppose that G is a finite group and H is a subgroup of G. H is said to be S-quasinormally embedded in G if for each prime p dividing the order of H, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G; H is said to be an SS-quasinormal subgroup of G if there is a subgroup B of G such that G= HB and H permutes with every Sylow subgroup of B. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1<| D|<| P| and study the structure of G under the assumption that every subgroup H of P with | H|=| D| is either S-quasinormally embedded or SS-quasinormal in G. Some recent results are generalized and unified. [ABSTRACT FROM AUTHOR]