Narrow Results

Suppose that $G$ is a finite group and $H\le K\le G$. We say that $H$ is called weakly characteristic in $K$ with respect to $G$ if for all subgroups $R$ of $K$ containing $H$, we have $H$ char $R$. In particular, when $H$ is a subgroup of prime power order and $K$ is a Sylow subgroup containing it, $H$ is simply said to be a weakly characteristic subgroup of $G$ or weakly characteristic in $G$. In this paper, we investigate the structure of finite groups by means of weakly characteristic subgroups.

Let G be a finite group and p a prime. We denote by ${\mathrm{\text{Irr}}}_p(G)$ the set of irreducible complex characters of G whose degrees are linear or divisible by p, and we write ${\mathrm{\text{R}}}_p(G)$ to denote the ratio of the sum of squares of irreducible character degrees in ${\mathrm{\text{Irr}}}_p(G)$ to the sum of irreducible character degrees in ${\mathrm{\text{Irr}}}_p(G)$. The Itô–Michler Theorem on character degrees states that ${\mathrm{\text{R}}}_p(G)=1$ if and only if G has a normal abelian Sylow p-subgroup. We generalize this theorem as follows: if ${\mathrm{\text{R}}}_p(G)<\frac{p+1}{2}$, then G has a normal Sylow p-subgroup.

Our main result here is the following theorem: Let G = AT, where A is a Hall π-subgroup of G and T is p-nilpotent for some prime p ∉ π, let P denote a Sylow p-subgroup of T and assume that A permutes with every Sylow subgroup of T. Suppose that there is a number p^k such that 1 < p^k < |P| and A permutes with every subgroup of P of order p^k and with every cyclic subgroup of P of order 4 (if p^k = 2 and P is non-abelian). Then G is p-supersoluble.

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.

Let G be a finite group and π_e(G) be the set of element orders of G. Let k ∈ π_e(G) and m_k be the number of elements of order k in G. Set nse(G) ≔ {m_k ∣ k ∈ π_e(G)}. In this paper, it is proved if G is a group with the following properties, then G ≅ PGL(2, p).

(1) p > 3 is prime divisor of ∣G∣ but p² does not divide ∣G∣.

(2) nse(G) = nse(PGL(2, p)).

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.

Let $P$ be a Sylow $p$-subgroup and $b(G)$ be the largest irreducible character degree of a finite nonabelian group $G$. Then $|P/O_p(G)|\le {(b{(G)}^p/p)}^{\frac{1}{p-1}}$.

In this paper, we prove that if every non-nilpotent maximal subgroup of a finite group $G$ has prime index then $G$ has a Sylow tower.

A subgroup $A$ of a group $G$ is called tcc-subgroup in $G$, if there is a subgroup $T$ of $G$ such that $G=AT$ and for any $X\le A$ and for any $Y\le T$, there exists an element $u\in 〈X,Y〉$ such that $X{Y}^u\le G$. The notation $H\le G$ means that $H$ is a subgroup of a group $G$. In this paper, we proved the supersolubility of a group $G=AB$ in the following cases: $A$ and $B$ are supersoluble tcc-subgroups in $G$; all Sylow subgroups of $A$ and of $B$ are tcc-subgroups in $G$; all maximal subgroups of $A$ and of $B$ are tcc-subgroups in $G$. Besides, the supersolubility of a group $G$ is obtained in each of the following cases: all maximal subgroups of every Sylow subgroup of $G$ are tcc-subgroups in $G$; every subgroup of prime order or 4 is a tcc-subgroup in $G$; all 2-maximal subgroups of $G$ are tcc-subgroups in $G$.

Denote by ${\nu }_p(G)$ the number of Sylow $p$-subgroups of $G$. For every subgroup $H$ of $G$, it is easy to see that ${\nu }_p(H)\le {\nu }_p(G)$, but ${\nu }_p(H)$ does not divide ${\nu }_p(G)$ in general. Following [W. Guo and E. P. Vdovin, Number of Sylow subgroups in finite groups, J. Group Theory 21(4) (2018) 695–712], we say that a group $G$ satisfies DivSyl(p) if ${\nu }_p(H)$ divides ${\nu }_p(G)$ for every subgroup $H$ of $G$. In this paper, we show that “almost for every” finite simple group $S$, there exists a prime $p$ such that $S$ does not satisfy DivSyl(p).

The norm $N(G)$ of a group $G$ is the intersection of the normalizers of all subgroups in $G$. In this paper, the norm is generalized by studying on Sylow subgroups and ${\mathscr{H}}$-subgroups in finite groups which is denoted by $C(G)$ and $A(G)$, respectively. It is proved that the generalized norms $A(G)$ and $C(G)$ are all equal to the hypercenter of $G$.

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 bcl(G) denote the largest conjugacy class length of a finite group G. In this note, we prove that if bcl(G)<p² for a prime p, then |G:O_p(G)|_p≤p.

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.

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 .

Let $𝔉$ be a hereditary saturated formation such that $𝔉\subseteq {𝔗}_{\prec }\cap {𝔑}^2$ (${𝔗}_{\prec }$ denotes the class of all Sylow tower groups of type $\prec$). In this paper, the concept of a fan of a subgroup of a group $G$, introduced in 1979 by Borevich, is used to prove that $G$ belongs to $𝔉$. We proved that a finite group $G\in 𝔉$ if and only if $\pi (G)\subseteq \pi (𝔉)$ and every basic subgroup of the fan of every Sylow subgroup in $G$ is $𝔉$-subnormal. We have also obtained that the group $G$ lies in $𝔉$ if and only if all the basic subgroups of the fans of all Sylow subgroups in $G$ lie in $𝔉$.