Narrow Results

Let $G$ be a finite group and $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes. A set ${\mathscr{H}}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of ${\mathscr{H}}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some ${\sigma }_i\in \sigma$ and ${\mathscr{H}}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma (G)$. A group $G$ is said to be a $\sigma$-full group if $G$ possesses a complete Hall $\sigma$-set. A subgroup $H$ of $G$ is said to be $s\sigma$-quasinormal in $G$ if there exists a $\sigma$-full subgroup $T$ of $G$ such that $G=HT$ and for all ${\sigma }_i\in \sigma (T)$, $H$ permutes with every Hall ${\sigma }_i$-subgroup of $T$. In this paper, we main investigate finite groups all of whose cyclic subgroups of order $2$ or $4$ are $s\sigma$-quasinormal subgroups. Some new criteria for $p$-nilpotency and supersolubilities are established.

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.

A subgroup $A$ of a finite group $G$ is called a local covering subgroup of $G$ if $A^G=AB$ for all maximal $G$-invariant subgroups $B$ of $A^G$. In this note, we characterize the finite groups $G$ in which all subgroups of order $p^d$ enjoy the local covering property, where $p^d$ is a prime power with $|G{|}_p\ge p^d\ge p$.

Let G be a finite group. A subgroup H of G is called weakly normal in G if H^g ≤ N_G(H) implies g ∈ N_G(H) for all g ∈ G. A finite group G is called an -group if all cyclic subgroups of G of order prime or 4 are weakly normal in G. In this paper, the structure of finite groups all of whose second maximal subgroups satisfy -property has been characterized.

A subgroup H of G is said to be self-conjugate-permutable if HH^x = H^xH implies H^x = H. A finite group G is called PSC-group if every cyclic subgroup of group G of prime order or order 4 is self-conjugate-permutable. In the paper, first we give the structure of finite group G, all of whose maximal subgroups are PSC-groups. Then we also classified that finite group G all of whose second maximal subgroups are PSC-groups.

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 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.

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 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 and A be finite groups of relative coprime orders and A act on G via automorphisms. In this paper, we prove that when every maximal A-invariant subgroup of G that contains the normalizer of some Sylow subgroup has prime index, then G is supersolvable; if every non-nilpotent maximal A-invariant subgroup of G has prime index or is normal in G, then G is a Sylow tower group.

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be a BNA-subgroup of $G$ if either $H^x=H$ or $x\in 〈H,H^x〉$ for all $x\in G$. A subgroup $H$ of $G$ is said to be a weakly BNA-subgroup of $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T$ is a BNA-subgroup of $G$. In this paper, we investigate the structure of a finite group $G$ under the assumption that every minimal subgroup of $G$ not having a supersolvable supplement in $G$ is a weakly BNA-subgroup of $G$.

Let $G$ be a finite group. A subgroup $H$ of $G$ is called Hall normally embedded in $G$ if $H$ is a Hall subgroup of the normal closure $H^G$. In this paper, we fix a subgroup $D$ of Sylow subgroup $P$ of $G$ with $1<\left|D\right|<\left|P\right|$ and study the structure of $G$ under the assumption that all subgroups $H$ of $P$ with $\left|H\right|=\left|D\right|$ are Hall normally embedded in $G$.

In this paper, we study the structure of finite groups G under the assumption that certain abelian subgroups of largest possible exponent of prime power order lie in the 𝔉-hypercenter of G. We improve and extend some results of Ezzat, Shaalan, Elashiry, Asaad, Ramadan, and Buckley.

Let be a saturated formation containing , and G be a finite group. Li etc. proposed a problem: whether there is a normality such that the following two statements are equivalent: (i) . (ii) There exists a normal subgroup H of G such that and for each Sylow subgroup P of F*(H), every member in some satisfies the above normality in G. In this paper, we find a normality satisfying the above problem. Moreover, by using the concept of ℋ-subgroups, we obtain other results about the influence of the members of some fixed on the structure of G.

Let H, C and D be subgroups of a finite group G. We call the pair (C, D) a θ*-pair for H if it satisfies the following conditions: (i) D = (C∩H)_G, 〈C^g, H〉 = G for every g ∈ G, (ii) KH < G for every K/D ◃ G/D with K/D < C/D. In this paper, we obtain several results on the maximal θ*-pair which implies G to be solvable, supersolvable and nilpotent.