Narrow Results

Let $G$ be a finite group. A subgroup $H$ of $G$ is called $s$-permutable in $G$ if $H$ permutes with every Sylow subgroup of $G.$ A subgroup $H$ of $G$ is called weakly $S\mathrm{\Phi }$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K\le \mathrm{\Phi }(H)H_{sG},$ where $\mathrm{\Phi }(H)$ is the Frattini subgroup of H and $H_{sG}$ is the subgroup of $H$ generated by all these subgroups of $H$ that are $s$-permutable in $G.$ Using this concept, some results for a group to be $p$-nilpotent and supersolvable are given. These results improve and extend some new and recent results in the literature.

Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HG_p = G_pH for any Sylow p-subgroup G_p of G; H is called weakly s-supplemented subgroup of G if there is a subgroup T of G such that G = HT and H ∩ T ≤ H_sG, where H_sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of minimal weakly s-supplemented subgroups on the structure of finite groups and generalize some recent results. Furthermore, we give a positive answer in the minimal subgroup case for Skiba's Open Questions in [On weakly s-permutable subgroups of finite groups, J. Algebra315 (2007) 192–209].

Let H be a subgroup of a group G. A subgroup H of G is said to be S-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. We say that H is weakly S-quasinormally embedded in G if there exists a normal subgroup T of G such that HT ⊴ G and H ∩ T is S-quasinormally embedded in G. In this paper, we investigate further the influence of weakly S-quasinormally embedded subgroups on the structure of finite groups. A series of known results are generalized.

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G, say G_p. Let C be a nonempty subset of G. A subgroup H of G is said to be C-ℨ-permutable (conjugate-ℨ-permutable) subgroup of G if there exists some x ∈ C such that H^xG_p = G_pH^x, for all G_p ∈ ℨ. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-ℨ-permutable (conjugate-ℨ-permutable) subgroups of G. Our results improve and generalize several results in the literature.

The paper considers the influence of Sylow normalizers, i.e. normalizers of Sylow subgroups, on the structure of finite groups. In the universe of finite soluble groups it is known that classes of groups with nilpotent Hall subgroups for given sets of primes are exactly the subgroup-closed saturated formations satisfying the following property: a group belongs to the class if and only if its Sylow normalizers do so. The paper analyzes the extension of this research to the universe of all finite groups.

Let H be a subgroup of a finite group G. We use H_sG to denote the S-core of H, that is, the subgroup of G generated by all those subgroups of H which are S-permutable in G. We say that H/H_sG is the S-cofactor of H. In this paper we study G under restrictions on the S-cofactors of some subgroups of G.

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G. A subgroup H of G is said to be ℨ-permutable of G if H permutes with every member of ℨ. A subgroup H of G is said to be a weakly ℨ-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ H_ℨ, where H_ℨ is the subgroup of H generated by all those subgroups of H which are ℨ-permutable subgroups of G. In this paper, we prove that if p is the smallest prime dividing the order of G and the maximal subgroups of G_p ∈ ℨ are weakly ℨ-permutable subgroups of G, then G is p-nilpotent. Moreover, we prove that if 𝔉 is a saturated formation containing the class of all supersolvable groups, then G ∈ 𝔉 iff there is a solvable normal subgroup H in G such that G/H ∈ 𝔉 and the maximal subgroups of the Sylow subgroups of the Fitting subgroup F(H) are weakly ℨ-permutable subgroups of G. These two results generalize and unify several results in the literature.

A subgroup H of a finite group G is called an -subgroup in G if there exists a normal subgroup T of G such that G = HT and H^g ∩ N_T(H) ≤ H for all g ∈ G. In this paper, we continue to investigate the structure of a finite group G with -subgroups. Some new results are given and many known results are generalized.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be an ${\mathscr{H}}C$-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^g\cap N_T(H)\le H$ for all $g\in G.$ We say that $H$ is weakly ${\mathscr{H}}C$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $H^G=HT$ and $H^g\cap N_T(H)\le H$ for all $g\in G$. In this paper, we investigate the structure of the finite group $G$ under the assumption that some subgroups of prime power order are weakly ${\mathscr{H}}C$-embedded in $G$. Our results improve and generalize several recent results in the literature.

A subgroup $H$ of a finite group $G$ is said to be a partial ${CAP}^{\ast }$-subgroup of $G$ if there exists a chief series ${\mathrm{\Gamma }}_H$ of $G$ such that $H$ either covers or avoids each non-Frattini chief factor of ${\mathrm{\Gamma }}_H$. In this paper, we study the influence of the partial ${CAP}^{\ast }$-subgroups on the structure of finite groups. Some new characterizations of the hypercyclically embedded subgroups, $p$-nilpotency and supersolubility of finite groups are obtained.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be a Hall $s$-semiembedded subgroup of $G$ if $H$ is a Hall subgroup of $〈H,P〉$ for any $P\in {\mathrm{\text{Syl}}}_p(G)$, where $(p,|H|)=1$. In this paper, we investigate the influence of Hall $s$-semiembedded subgroups on the structure of the finite group $G$. Some new results about $G$ to be a $𝔉$-group are obtained, where $𝔉$ is a saturated formation.

A subgroup $H$ of a group $G$ is said to be an $\mathcal{\mathscr{H}}C$-subgroup of $G$, if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H^g\cap N_K(H)\le H$, for all $g\in G$. In this paper, we investigate the structure of groups based on the assumption that every subgroup of $P\cap G^{{{𝒩}}_p}$ of order $p$ or 4 (if $p=2$) is an $\mathcal{\mathscr{H}}C$-subgroup of $N_G(P)$, here $P$ is a Sylow $p$-subgroup of $G$. Some results for a group to be $p$-nilpotent and supersolvable are obtained and many known results are generalized.

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an ${\mathscr{H}}$-subgroup of $G$ if $N_G(H)\cap H^g\le H$ for all $g\in G$. We say that $H$ is weakly ${\mathscr{H}}{𝒞}$-embedded in $G$ if $G$ has a normal subgroup $T$ such that $H^G=HT$ and $N_T(H)\cap H^g\le H$ for all $g\in G,$ where $H^G$ is the normal closure of $H$ in $G$. For each prime $p$ dividing the order of $G,$ let $P$ be a Sylow $p$-subgroup of $G$. We fix a subgroup of $P$ of order $d$ with $1<d<\left|P\right|$ and study the structure of $G$ under the assumption that every subgroup of $P$ of order $p^{n}d$$(n=0,1)$ is weakly ${\mathscr{H}}{𝒞}$-embedded in $G$. Our results improve and generalize several recent results in the literature.

Let $G$ be a finite group. A subgroup $H$ of $G$ is called to be $S$-permutable in $G$ if $H$ permutes with all Sylow subgroups of $G$. A subgroup $H$ of $G$ is said to be $SS$-supplemented in $G$ if there exists a subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is $S$-permutable in $K$. In this paper, we investigate $p$-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

Let $G$ be a group. A subgroup $H$ of $G$ is called an ${\mathscr{H}}$-subgroup in $G$ if $N_G(H)\cap H^x\le H$ for all $x\in G$. Furthermore, a subgroup $H$ of $G$ is called a weakly ${\mathscr{H}}$-subgroup in $G$ if there exists a normal subgroup $K$ of $G$ such that $G=HK$ and $H\cap K$ is an ${\mathscr{H}}$-subgroup in $G$. In this paper, some new criteria for a group to be $p$-nilpotent and supersolvable are given.

Let G be a finite group. We say that a subgroup H of G is -normal in G if G has a subnormal subgroup T such that TH = G and (H ∩ T)H_G/H_G is contained in the -hypercenter of G/H_G, where is the class of the finite supersoluble groups. We study the structure of G under the assumption that some subgroups of G are -normal in G.

It is proved that the lattice of totally saturated formations of finite groups is distributive. Thus, we give an affirmative answer to the problem proposed by Shemetkov, Skiba and Guo.

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set of maximal subgroups of P such that . It is shown that if every member of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

Suppose that G is a finite group and H is a subgroup of G. We say that H is s-semipermutable in G if HG_p = G_pH for any Sylow p-subgroup G_p of G with (p, |H|) = 1; H is s-semiembedded in G if there is a normal subgroup T of G such that HT is s-semipermutable in G and H ∩ T ≤ H_ssG, where H_ssG is the subgroup of H generated by all those subgroups of H which are s-semipermutable in G. We investigate the influence of s-semiembedded subgroups on the structure of finite groups. Some recent results are generalized and unified.

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let H_sG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ H_sG.

Our main result here is the following theorem. Let be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that . Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then .