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.

Let $G$ be a finite group. How minimal subgroups can be embedded in $G$ is a question of particular interest in studying the structure of $G$. A subgroup $H$ of $G$ is called $s$-permutable in $G$ if $HP=PH$ for all Sylow subgroups $P$ of $G$. A subgroup $H$ of $G$ is called $n$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $H^G=HT$ and $H\cap T\le H_{sG}$, where $H_{sG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutable in $G$. In this paper, we investigate the structure of the finite group $G$ with $n$-embedded subgroups.

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.

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.

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.

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.

In this note, we use fewer $p$-subgroups $H$ with the condition $O^p(G)\cap H⊴O^p(G)$ to investigate the structure of finite groups. We prove that for a fixed prime $p$, a given Sylow $p$-subgroup $P$ of a finite group $G$, and a power $d$ of $p$ dividing $\left|G\right|$ such that $p^2\le d<\left|P\right|$, if $H\cap O^p(G)$ is normal in $O^p(G)$ for all non-cyclic normal subgroups $H$ of $P$ with $\left|H\right|=d$, then either $G$ is $p$-supersoluble or else $|P\cap O^p(G)|>d$. This extends the main result of Guo and Isaacs (Conditions on $p$-subgroups implying $p$-nilpotence or $p$-supersolvability, Arch. Math.105 (2015) 215–222). We also derive some applications of the above result which extend some known results.

Suppose that $G$ is a finite group and $H$ is a 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 $B$. In this note, we fix in every non-cyclic Sylow subgroup $P$ of $G$ some subgroup $D$ satisfying $1<\left|D\right|<\left|P\right|$ and study the $p$-nilpotency of $G$ under the assumption that every subgroup $H$ of $P$ with $\left|H\right|=\left|D\right|$ is $SS$-quasinormal in $G$. The Frobenius theorem is 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 $g\in G$; $H$ is called 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 $G$. In this paper, we study the $p$-nilpotence of a group $G$ in which every subgroup of order $d$ of a Sylow $p$-subgroup $P$ with $1<d<\left|P\right|$ is weakly ${\mathscr{H}}C$-embedded in $G$. Many recent results in the literature related to $p$-nilpotence of $G$ are generalized.

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 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 investigate the structure of a finite group $G$ under the assumption that certain subgroups of prime power order are Hall normally embedded in $G$.

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. 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 $G$.

The norm N(G) of a group G is the intersection of normalizers of all the subgroups of G. Let G be a finite group, p a prime dividing the order of G, and P a Sylow p-subgroup of G. In this paper, it is proved that G is p-nilpotent if Ω₁(P) ≤ N(N_G(P)), and when p=2, . Some applications of this result are given. Finally, a class of finite p-groups in which the index of the norm is exactly p is described.

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.

We prove that a finite group G is p-supersolvable or p-nilpotent if some subgroups of G are weakly s-semipermutable in G. Several earlier results are generalized.