Search

Let σ = {σi|i ∈ I} be some partition of the set of all primes ℙ, G 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 member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-semipermutable in G with respect to H if HHix = HixH for all x ∈ G and all Hi ∈ H such that (|H|, |Hi|) = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G. [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]

A subgroup H of a finite group G is said to be an H-subgroup in G if NG(H)∩Hg≤H for all g∈G; we say that H is an HC-subgroup in G if there is a normal subgroup T of G such that G=HT and Hg∩NT(G)≤H for all g∈G. In this note, we obtain some interesting results by applying HC-subgroups. As applications, many known results are generalized in recent literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give answers to some questions concerning intersections of system normalisers and prefrattini subgroups of finite soluble groups raised by the third author, Shemetkov and Vasil'ev in the Kourovka Notebook [10] . Our approach depends on results on regular orbits and it can be also used to extend a result of Mann [9] concerning intersections of injectors associated to Fitting classes. [ABSTRACT FROM AUTHOR]

For any a , b , c , n ∈ N with 1 < a , b , c ≤ n − 2 , then there exist α , β ∈ S n such that o ( α ) = a , o ( β ) = b and o ( α β ) = c . This is a conjecture of Stefan Kohl and which is closely related to problem on covers of the complex projective line. In this note we prove the conjecture is true. [ABSTRACT FROM AUTHOR]

H is called an ℳ-embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H ∈ Syl( B) and B is ℳ-supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ℳ-embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ⩽ | P| and d divides | P|. If every subgroup H of P with | H| = d is ℳ-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/ C is cyclic of order 5, (2) I/C is 5′-group, (3) I/ C ≅ A . [ABSTRACT FROM AUTHOR]

A subgroup H of G is called complemented in G if there exists a subgroup K of G such that G = H K and H ∩ K = 1 . The aim of this paper is to prove the following: A finite group G is solvable if and only if its Sylow 3-, 5- and 7-subgroups are complemented in G . [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]

A formation 𝔉 of finite groups has the generalized Wielandt property for residuals, or 𝔉 is a GWP-formation, if the 𝔉-residual of a group generated by two 𝔉-subnormal subgroups is the subgroup generated by their 𝔉-residuals. The main aim of the paper is to determine some sufficient conditions for a finite group to split over its 𝔉-residual. [ABSTRACT FROM PUBLISHER]

A subgroupHof a finite groupGis calledc*-supplemented inGif there exists a subgroupKofGsuch thatG = HKandH∩KisS-quasinormally embedded inG. In this paper, we investigate the localc*-supplementation of maximal subgroups of some Sylowp-subgroup and present some sufficient and necessary conditions for a finite group to bep-nilpotent. As applications, we give some sufficient conditions for a finite group to be in a saturated formation. [ABSTRACT FROM PUBLISHER]

An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow p-subgroups, where p is the defining characteristic. [ABSTRACT FROM AUTHOR]

In this note, we give a new brief proof of a theorem in the article "Sylow Normalizers and p-Nilpotence of Finite Groups" by Zhang, Li, and Miao [9]. [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]

Let ℱ be a subgroup-closed formation and letGbe a finite group. Then we definiteNℱ(G) as the intersection of the normalizers of the ℱ-residuals of all subgroups ofG. The main aim of this article is to show the influence of the embedding of this subgroup in the group structure. [ABSTRACT FROM PUBLISHER]

A subgroupHof a finite groupGis said to satisfy Π-property inGif for every chief factorL/KofG, |G/K:NG/K(HK/K∩L/K)| is a π(HK/K∩L/K)-number. A subgroupHofGis called Π-supplemented inGif there exists a subgroupTofGsuch thatG = HTandH∩T ≤ I ≤ H, whereIsatisfies Π-property inG. In this article, we investigate the structure of a finite groupGunder the assumption that some primary subgroups ofGare Π-supplemented inG. The main result we proved improves a large number of earlier results. [ABSTRACT FROM AUTHOR]

We show that there is no absolute bound on the Fitting height of a group with two Sylow numbers. [ABSTRACT FROM AUTHOR]

The major aim of the note is to strengthen the results of Asaad inCommunications in Algebra, 42 (2014), 2319–2330. [ABSTRACT FROM AUTHOR]

Let π be a set of primes. By H. Wielandt's definition, Sylow π-theorem holds for a finite group G if all maximal π -subgroups of G are conjugate. In the paper, the following statement is proven. Assume that π is a union of disjoint subsets σ and τ and a finite group G possesses a π -Hall subgroup which is a direct product of a σ -subgroup and a τ -subgroup. Furthermore, assume that both the Sylow σ -theorem and τ -theorem hold for G . Then the Sylow π -theorem holds for G . This result confirms a conjecture posed by H. Wielandt in 1959. [ABSTRACT FROM AUTHOR]

A subgroupAof a finite groupGis called aTI-subgroup if eitherA∩Ax = 1 orA∩Ax = Aholds for allx ∈ G. In this paper, finite group all of whose meta-cyclic subgroups areTI-subgroups are classified completely. In particular, such groups are solvable. [ABSTRACT FROM PUBLISHER]

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 : N G ( 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 : N G ( 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 H T = G and H ∩ T ⊆ K Z F ( T ) ; H is F τ -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 / H G and H ¯ = H / H G . In this paper we study the structure of a group G under the condition that some primary subgroups of G are F τ -supplemented in G . In particular, we prove the following result. Theorem A. Let F be a saturated formation containing the class U of all supersoluble groups, E a normal subgroup of G with G / E ∈ F , X = E or 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 U τ -supplemented in G, then G ∈ 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. [ABSTRACT FROM AUTHOR]

Given a chief factor H / K of a finite group G , we say that a subgroup A of G avoids H / K if H ∩ A = K ∩ A ; if H A = K A , then we say that A covers H / K . If A either covers or avoids the chief factors of some given chief series of G , we say that A is a partial CAP-subgroup of G . Assume that G has a Sylow p -subgroup of order exceeding p k . If every subgroup of order p k , where k ≥ 1 , and every subgroup of order 4 (when p k = 2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G , then G is p -soluble of p -length at most 1. [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]

We show that every saturated fusion system F has a unique minimal F -characteristic biset Λ F . We examine the relationship of Λ F with other concepts in p -local finite group theory: In the case of a constrained fusion system, the model for the fusion system is the minimal F -characteristic biset, and more generally, any centric linking system can be identified with the F -centric part of Λ F as bisets. We explore the grouplike properties of Λ F , and conjecture an identification of normalizer subsystems of F with subbisets of Λ F . [ABSTRACT FROM AUTHOR]

Let $$H$$ be a subgroup of a finite group $$G$$ . Then we say that $$H$$ is $$S$$ -propermutable in $$G$$ provided $$G$$ has a subgroup $$B$$ such that $$G=N_{G}(H)B$$ and $$H$$ permutes with all Sylow subgroups of $$B$$ . In this paper we analyze the influence of $$S$$ -propermutable subgroups on the structure of $$G$$ . [ABSTRACT FROM AUTHOR]

Let H be a subgroup of a finite group G. H is nearly SS-embedded in G if there exists an S-quasinormal subgroup K of G, such that HK is S-quasinormal in G and H ∩ K ≤ H, where H is the subgroup of H, generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, the authors investigate the influence of nearly SS-embedded subgroups on the structure of finite groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group. A subgroup H of G is called an NE-subgroup of G if it satisfies H∩ N( H)= H. A subgroup H of G is said to be a N E-subgroup of G if there exists a subnormal subgroup T of G such that G= H T and H∩ T is a NE-subgroup of G. In this article, we investigate the structure of G under the assumption that subgroups of prime order are N E-subgroups of G. The finite groups, all of whose minimal subgroups of the generalized Fitting subgroup are N E-subgroups are classified. [ABSTRACT FROM AUTHOR]

A subgroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ of a finite group $G$ is said to be S-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $H $. A finite group $G$ is an MS-group if the maximal subgroups of all the Sylow subgroups of $G$ are S-semipermutable in $G$. The aim of the present paper is to characterise the finite MS-groups. [ABSTRACT FROM AUTHOR]

In this paper, we characterize finite groupsGsatisfying that, for every prime power divisorpnof the order ofG, there exists a subgroupHof indexpn(of orderpn) inGsuch thatHis normal or abnormal inG. [ABSTRACT FROM AUTHOR]

We give some conditions under which a finite group belongs to a Baer-local formation containing all finite supersoluble groups. [ABSTRACT FROM AUTHOR]

We show that if an inclusion of finite groups H≤ G of index prime to p induces a homeomorphism of mod p cohomology varieties, or equivalently an F-isomorphism in mod p cohomology, then H controls p-fusion in G, if p is odd. This generalizes classical results of Quillen who proved this when H is a Sylow p-subgroup, and furthermore implies a hitherto difficult result of Mislin about cohomology isomorphisms. For p=2 we give analogous results, at the cost of replacing mod p cohomology with higher chromatic cohomology theories. The results are consequences of a general algebraic theorem we prove, that says that isomorphisms between p-fusion systems over the same finite p-group are detected on elementary abelian p-groups if p odd and abelian 2-groups of exponent at most 4 if p=2. [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]

LetGbe a finite group. A subgroupHofGis said to be weaklys-supplemented inGif there exists a subgroupKofGsuch thatG = HKandH∩K ≤ Hs G, whereHs Gis the subgroup ofHgenerated by all those subgroups ofHwhich ares-quasinormal inG. In this article, we investigate the structure ofGunder the assumption that some families of subgroups ofGare weaklys-supplemented inG. Some recent results are generalized. [ABSTRACT FROM PUBLISHER]

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]

Let G be a finite group and let x denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G. [ABSTRACT FROM AUTHOR]

In this article, we prove a conjecture of J. G. Thompson for the finite simple group2Dn(q). More precisely, we show that every finite groupGwith the propertyZ(G) = 1 andN(G) = N(2Dn(q)) is necessarily isomorphic to2Dn(q). Note thatN(G) is the set of lengths of conjugacy classes ofG. [ABSTRACT FROM PUBLISHER]

Let ℱ be a class of groups andGa finite group. We call a set Σ of subgroups ofGaG-covering subgroup system for ℱ ifG∈ ℱ whenever Σ ⊆ ℱ. For a non-identity subgroupHofG, we put ΣHbe some set of subgroups ofGwhich contains at least one supplement inGof each maximal subgroup ofH. Letp ≠ qbe primes dividing |G|,P, andQbe non-identity ap-subgroup and aq-subgroup ofG, respectively. We prove that ΣPand ΣP ∪ ΣQareG-covering subgroup systems for many classes of finite groups. [ABSTRACT FROM AUTHOR]

A subgroupHofGis called ℳp-supplemented inGif there exists a subgroupBofGsuch thatG = HBandTB < Gfor every maximal subgroupTofHwith |H:T| =pα. In this paper, we investigate the influence of ℳp-supplemented subgroup and some conditions forp-nilpotency andp-supersolvability of finite groups are obtained. [ABSTRACT FROM AUTHOR]

Baer and Wielandt in 1934 and 1958, respectively, considered that the intersection of the normalizers of all subgroups ofGand the intersection of the normalizers of all subnormal subgroups ofG. In this article, for a finite groupG, we define the subgroupS(G) to be intersection of the normalizers of all non-cyclic subgroups ofG. Groups whose noncyclic subgroups are normal are studied in this article, as well as groups in which all noncyclic subgroups are normalized by all minimal subgroups. In particular, we extend the results of Passman, Bozikov, and Janko to non-nilpotent finite groups. [ABSTRACT FROM AUTHOR]

A finite groupGis called anMNP-group if all maximal subgroups of every Sylow subgroup ofGare normal inG. In this article, we give a complete classification of those groups which are notMNP-groups but all of whose proper subgroups areMNP-groups. [ABSTRACT FROM AUTHOR]

Let H he a subgroup of a finite group G. H is stiiil to he k-supplemented in G If G has a subgroup T such that G -- HT and H ∩ T ≤ HSE, where HSE denotes the subgroup of H generated by all those subgroups of H, which are S-quasinormally embedded in G. In this article, some results about the γ-supplemented subgroups are obtained, by which we determine the structure of some classes of finite groups. In particular, some new characterizations of p-supersoluhility of finite groups are given under the assumption that some primary subgroups are γ-supplemented. As applications, a number of previous known results are generalized. [ABSTRACT FROM AUTHOR]

A subgroup H of a finite group G is called a QTI-subgroup if C G (x) ≤ N G (H) for any 1 ≠ x ∈ H. In this article, the finite groups all of whose second maximal subgroup are QTI-subgroups are classified. [ABSTRACT FROM PUBLISHER]

Let G be a finite group and H ≤ G. The subgroup H is called: S-permutable in G if HP = PH for all Sylow subgroups P of G; S-permutably embedded in G if each Sylow subgroup of H is also a Sylow subgroup of some S-permutable subgroup of G. Let H be a subgroup of a group G. Then we say that H is SQ-supplemented in G if G has a subgroup T and an S-permutably embedded subgroup C ≤ H such that HT = G and T ∩ H ≤ C. We study the structure of G under the assumption that some subgroups of G are SQ-supplemented in G. Some known results are generalized. [ABSTRACT FROM PUBLISHER]

Let G he a finite group. A subgroup H of G is called an H -subgroup in G if NG(H) ∩ Hx &les; H for all x ∈ G. A subgroup H of G is called weakly H-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩ K is an H -subgroup in G. In this article, we investigate the structure of the finite group G under the assumption that all maximal subgroups of every Sylow subgroup of some normal subgroup of G are weakly H-subgroups in G. Some recent results are extended and generalized. [ABSTRACT FROM AUTHOR]

A subgroup H of a group G is said to be an ℋC-subgroup of G if there exists a normal subgroup T of G such that G = HT and Hg ∩ NT(H) &les; H for all g ∈ G. In this article, some new characterizations for p-nilpoteney and supersolviibility of finite groups are presented. In addition, we discuss the structure of ℋC*-group ( a group G is called an ℋC*-group if every subgroup of G is an ℋC-subgroup of G) and prove that a group G is an ℋC*-group if and only if there is a nilpotent normal subgroup H of G such that G/H is abelian and each element of G induces a power automorphism on H by conjugation. [ABSTRACT FROM AUTHOR]

Let G be a finite group and ℱ a formation. A subgroup H is called ℱ-quasinormal in G if there exists a quasinormal subgroup T of G such that HT is quasinormal in G and (H ∩ T)H G /H G is contained in the ℱ-hypercenter of G/H G . In this article, we study the structure of finite groups by using ℱ-quasinormal subgroups and prove that: Let ℱ be a saturated formation containing 𝒰 and G be a group with a normal subgroup H such that G/H ∈ ℱ. If every maximal subgroup of every noncyclic Sylow subgroup of F*(H) having no supersolvable supplement in G is 𝒰-quasinormal in G, then G ∈ ℱ. [ABSTRACT FROM PUBLISHER]

A subgroup H of a group G is called a CAP*-subgroup of G, if H either covers or avoids every non-Frattini chief factor of G. In this article, we give some interesting properties of CAP*-subgroups. Furthermore, we determine the structure of finite groups based on the assumption that some subgroups are CAP*-subgroups and generalize some known results. [ABSTRACT FROM AUTHOR]

Let G = NwrCn be the wreath product of N by Cn, where N is a finite nilpotent group and Cn is a cyclic group of order n. Then the normalizer property holds for G. [ABSTRACT FROM AUTHOR]

Let G be a finite group and F a formation of finite groups. We say that a subgroup H of G is Fs-quasinormal in G if there exists a normal subgroup T of G such that HT is s-permutable in G and (H ∩ T)HG/HG is contained in the F-hypercenter [image omitted] of G/HG. In this article, we use Fs-quasinormal subgroups to study the structure of finite groups. A series of known results are generalized. [ABSTRACT FROM AUTHOR]

We call a subgroup H of a group G nearly s-normal in G if there exists N ⊴ G such that HN ⊴ G and H ∩ N ≤ HsG, where HsG is the largest s-permutable subgroup of G contained in H. In this article, we obtain some results about the nearly s-normal subgroups and use them to characterize the structure of finite groups. [ABSTRACT FROM AUTHOR]

Let G be a finite group and H, K two nontrivial subgroups of G. We say that G is a mutually m-permutable product of H and K if G = HK and every maximal subgroup of H permutes with K and every maximal subgroup of K permutes with H. We prove the following result: Let G = G1G2...Gr be a finite group such that G1, G2,...Gr are pairwise permutable subgroups of G and GiGj is a mutually m-permutable product for all i and j with i ≠ j. If the Sylow subgroups of all Gi are cyclic, then G is supersolvable. [ABSTRACT FROM AUTHOR]