Narrow Results

A subgroup $H$ of a group $G$ is called an $\mathcal{𝒮}\mathcal{𝒮}\mathcal{\mathscr{H}}$-subgroup in $G$ if there exists an $s$-permutable subgroup $K$ of $G$ such that $H^{sG}=HK$ and $H^g\cap N_K(H)\le H$ for all $g\in G$, where $H^{sG}$ is the intersection of all $s$-permutable subgroup of $G$ containing $H$. In this paper, we present two new sufficient and necessary conditions on the $p$-nilpotency of finite groups by use of a small quantity of $\mathcal{𝒮}\mathcal{𝒮}\mathcal{\mathscr{H}}$-subgroups of Sylow $p$-subgroups. A number of known results are improved and extended.

In this paper, we prove the p-nilpotency of a finite group G with the assumption that some subgroups of a Sylow subgroup are semi CAP-or c-supplemented subgroups. We get some interesting new results.

A subgroup H of a finite group G is said to be SS-quasinormal in G if there exists a supplement B of H in G such that H is permutable with every Sylow subgroup of B. In this paper, we get some new characterizations of supersolvability and p-nilpotency of G by assuming some subgroups of prime power order of G are SS-quasinormal.

A subgroup H of a group G is called Φ-s-supplemented in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Φ (H), where Φ(H) is the Frattini subgroup of H. We investigate the influence of Φ-s-supplemented subgroups on the p-nilpotency, p-supersolvability and supersolvability of finite groups.

A new criteria for the solvability of finite groups is obtained. It is proven that a finite group G is always solvable if

(i) H is subnormal in G or |N_G(H) : H| ≤ 2 for every non-cyclic 2-subgroup H of G that is not a subgroup of Z(G), or

(ii) H is subnormal in G or G is A₄-free and |N_G(H) : H| ≤ 6 for every non-cyclic 2-subgroup H of G that is not a subgroup of Z(G).

Moreover, some new criteria for a finite group being p-nilpotent are also obtained.

Consider a finite group G. A subgroup H of G is said to be E-supplemented in G if there is a subgroup T of G such that G = HT and H ∩ T ≤ H_eG, where H_eG denotes the subgroup of H generated by all those subgroups of H which are S-quasinormally embedded in G. In this paper, we investigate the influence of E-supplemented subgroups on the structure of finite 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, 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 p be a fixed prime, G a finite group and P a Sylow p-subgroup of G. The main results of this paper are as follows:

(1) If gcd(p-1, |G|) = 1 and p² does not divide |x^G| for any p′-element x of prime power order, then G is a solvable p-nilpotent group and a Sylow p-subgroup of G/O_p(G) is elementary abelian.

(2) Suppose that G is p-solvable. If p^p-1 does not divide |x^G| for any element x of prime power order, then the p-length of G is at most one.

(3) Suppose that G is p-solvable. If p^p-1 does not divide χ(1) for any χ ∈ Irr(G), then both the p-length and p′-length of G are at most 2.

Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is an ${\mathscr{H}}$-subgroup in $G$ if $N_G(H)\cap H\le H^g$ for all $g\in G$; $H$ is called weakly ${\mathscr{H}}$-subgroup in $G$ if it has a normal subgroup $K$ such that $G=HK$ and $H\cap K$ is an ${\mathscr{H}}$-subgroup in $G$. In this paper, we present some sufficient conditions for a group to be $p$-nilpotent under the assumption that certain subgroups of fixed prime power orders are weakly ${\mathscr{H}}$-subgroups in $G$. The main results improve and extend new and recent results in the literature.

Let $G$ be a finite group, $p$ a prime, $P$ a Sylow $p$-subgroup of $G$ and $d$ a power of $p$ such that $1<d<\left|P\right|$. Let $G_p^{\ast }$ denote the unique smallest normal subgroup of $G$ for which the corresponding factor group is abelian of exponent dividing $p-1$. Let ${𝔉}_1$, ${𝔉}_2$, ${𝔉}_3$ be classes of all $p$-groups, $p$-nilpotent groups and $p$-supersolvable groups, respectively, $G^{𝔉}$ be the $𝔉$-residual of $G$. Let $X\in \{{(G_p^{\ast })}^{{𝔉}_1},G^{{𝔉}_2},G^{{𝔉}_3}\}$. A subgroup $H$ of a finite group $G$ is said to have $\mathrm{\Pi }$-property in $G$, if for any $G$-chief factor $L/K$, $|G/K:N_{G/K}((H\cap L)K/K)|$ is a $\pi ((H\cap L)K/K)$-number. A normal subgroup $E$ of $G$ is said to be $p$-hypercyclically embedded in $G$ if every $p$-$G$-chief factor of $E$ is cyclic, where $p$ is a fixed prime. In this paper, we prove that $E$ is $p$-hypercyclically embedded in $G$ if and only if for some $p$-subgroups $H$ of $E$, $H\cap X$ have $\mathrm{\Pi }$-property in $G$.

Let $G$ be a finite group and $H$ a subgroup of $G$. $H$ is said to be $NH$-embedded in $G$ if there exists a normal subgroup $T$ of $G$ such that $HT$ is a Hall subgroup of $G$ and $H\cap T\le H_{\bar{s}G}$, where $H_{\bar{s}G}$ is the largest $s$-semipermutable subgroup of $G$ contained in $H$. In this paper, we give some new characterizations of $p$-nilpotent and supersolvable groups by using $NH$-embedded subgroups. Some known results are generalized.

Let $G$ and $A$ be groups where $A$ acts on $G$ by automorphisms. We say “the action of $A$ on $G$ is good” if the equality $H=[H,B]C_H(B)$ holds for any subgroup $B$ of $A$ and for any $B$-invariant subgroup $H$ of $G$. It is straightforward that every coprime action is a good action. In this work, we extend some results due to Ward, Gross, Shumyatsky, Jabara and Meng and Guo under coprime action to good action.

Let $H\le K$ be subgroups of a finite group $G$, then $H$ is called strongly closed in $K$ with respect to $G$ if $H^g\cap K\le H$ for every $g\in G$, and in particular, $H$ is simply called strongly closed in $G$ if $H$ is strongly closed in $N_G(H)$ with respect to $G$. Let $H$ be a subgroup of a finite group $G$, then $H$ is called Hall $s$-semiembedded in $G$ if $H$ is a Hall subgroup of $〈H,P〉$ for every $P\in {\mathrm{\text{Syl}}}_p(G)$, where $(|H|,p)=1$. In this paper, we obtain some criteria for $p$-nilpotency of a finite group and extend some known results concerning strongly closed and Hall $s$-semiembedded subgroups. In particular, we generalize some main results of Guo and Li [Hall $s$-semiembedded subgroups and $p$-nilpotency of finite groups, Southeast Asian Bull. Math. 42(3) (2018) 367–374] and Kong [New characterizations of $p$-nilpotency of finite groups, J. Algebra Appl. 20(11) (2021) Article ID: 2150215, 6 pp.].

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, we prove that if every non-nilpotent maximal $A$-invariant subgroup of $G$ is normal, then $G$ is $p$-nilpotent or $p$-closed for each prime divisor $p$ of $\left|G\right|$, which improves a theorem obtained by Beltr$́a$n and Shao.

A subgroup H of a finite group G is called an ℋ-subgroup of G if N_G(H) ∩ H^g ≤ H for all g ∈ G. The set of all ℋ-subgroups of a finite group G is denoted by ℋ(G). In this paper, a sufficient condition about p-nilpotency is given and some new results for a finite group G to be p-nilpotent or supersolvable are obtained based on the assumption that some subgroups belong to ℋ(G).