Narrow Results

A well-known theorem of Huppert states that a finite group is soluble if its every proper subgroup is supersoluble. In this paper, we proved the following result: let G be a finite group. (1) If G has exactly n non-supersoluble proper subgroups, where 0 ≤ n ≤ 7 and n ≠ 5, then G is soluble. (2) G is a non-soluble group with exactly five non-supersoluble proper subgroups if and only if all non-supersoluble proper subgroups are conjugate maximal subgroups and G/Φ(G) ≅ A₅, where Φ(G) is the Frattini subgroup of G. Furthermore, we also considered the influence of the number of non-abelian proper subgroups on the solubility of finite groups.

We study groups having the property that every non-cyclic subgroup contains its centralizer. The structure of nilpotent and supersolvable groups in this class is described. We also classify finite p-groups and finite simple groups with the above defined property.

In this paper, we study a finite group $G$ such that $|H:H_G|$ is a prime for each non-normal subgroup $H$ of $G$. We prove that such a group must contain a big abelian subgroup. More specifically, if such a group $G$ is not supersoluble, then there is an abelian subgroup $A$ such that $|G:A||2pq$, and if $G$ is supersoluble, then there is an abelian subgroup $A$ such that $|G:A||2^5$ or $|G:A||p^2{q}^2$, where $p$ and $q$ are primes.

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.

The subgroup structure of a finite group, under the assumption that its every non-nilpotent maximal subgroup has prime index, is studied in the paper.

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 $G$.

Let $\sigma =\{{\sigma }_i|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $\sigma (G)=\{{\sigma }_i|{\sigma }_i\cap \pi (|G|)\ne \varnothing \}$. A set ${\mathscr{H}}$ of subgroups of $G$ is said to be a complete Hall$\sigma$-set of $G$ if every non-identity member of ${\mathscr{H}}$ is a Hall ${\sigma }_i$-subgroup of $G$ for some $i\in I$ and ${\mathscr{H}}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma (G)$. A subgroup $H$ of $G$ is said to be: $\sigma$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set ${\mathscr{H}}$ such that $H{A}^g=A^{g}H$ for all $A\in {\mathscr{H}}$ and all $g\in G$; $\sigma$-semipermutable in $G$ if $G$ has a complete Hall $\sigma$-set ${\mathscr{H}}$ such that $H{A}^g=A^{g}H$ for all $g\in G$ and all ${\sigma }_i$-group $A\in {\mathscr{H}}$ with ${\sigma }_i\in \sigma (G)\setminus \sigma (H)$. We say that $H$ is $\sigma$-semiembedded in $G$ if there exists a $\sigma$-permutable subgroup $T$ of $G$ such that $HT$ is a $\sigma$-permutable subgroup of $G$ and $H\cap T\le H_{\sigma sG}$, where $H_{\sigma sG}$ denotes the subgroup of $H$ which is generated by all $\sigma$-semipermutable subgroups of $G$ contained in $H$. In this paper, we study the influence of $\sigma$-semiembedded subgroups on the structure of finite groups. Some known results are generalized and unified.

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.

It is proved that a finite group G=AB is supersoluble provided that A and B are such supersolube subgroups of G that every cyclic subgroup of A permutes with every Sylow subgroup of B and if in return every cyclic subgroup of B permutes with every Sylow subgroup of A.

Let H and T be subgroups of a finite group G. We say that H is completely c-permutable with T in G if there exists an element x ∈ 〈H,T〉 such that HT^x = T^xH. In this paper, we use this concept to determine the supersolubility of a group G = AB, where A and B are supersoluble subgroups of G. Some criterions of supersolubility of such groups are obtained and some known results are generalized.

The structure of groups in which many subgroups have a certain property χ has been investigated for several choices of the property χ. In particular, groups whose non-normal subgroups are supersoluble are studied in this paper. Moreover, groups with only finitely many normalizers of non-supersoluble groups are considered.

Let G be a finite group, $\sigma =\left\{{\sigma }_i\right|i\in I\}$ be some partition of the set of all primes and $\sigma (G)=\left\{{\sigma }_i\right|{\sigma }_i\cap \pi (G)\ne \varnothing \}$. We say that a subgroup H of G is nearly σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT is a σ-permutable subgroup of G and $H\cap T\le H_{\sigma eG}$, where H_σeG is the subgroup of H generated by all those subgroups of H which are σ-permutably embedded in G. In this paper, we study the structure of G under the condition that some given subgroups of G are nearly σ-embedded in G. Some known results are generalized.

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 G be a finite group and denote a formation of finite groups by . We call a subgroup H of G-supplemented in G if there exists a normal subgroup K of G such that G = HK and (H ∩ K)H_G/H_G is contained in the -hypercenter of G/H_G. In this paper, we use -supplemented subgroups to study the structure of finite groups. Some known results in the literature are generalized.

A subgroup H of a group G is said to be K-ℙ-subnormal inG [A. F. Vasilyev, T. I. Vasilyeva and V. N. Tyutyanov, On finite groups with almost all K-ℙ-subnormal Sylow subgroups, in Algebra and Combinatorics: Abstracts of Reports of the International Conference on Algebra and Combinatorics on Occasion the 60th Year Anniversary of A. A. Makhnev (Ekaterinburg, 2013), pp. 19–20] if there exists a chain of subgroups H = H₀ ≤ H₁ ≤ ⋯ ≤ H_n = G such that either H_i-1 is normal in H_i or |H_i : H_i-1| is a prime, for i = 1, …, n. In this paper, we describe finite groups in which every second maximal subgroup is K-ℙ-subnormal.

In this paper, the classes of groups with given systems of $𝔉$-subnormal subgroups are studied. In particular, it is showed that if $ℌ$ and $𝔉$ are a saturated homomorph and a hereditary saturated formation, respectively, then the class of groups whose $ℌ$-subgroups are all $𝔉$-subnormal is a hereditary saturated formation. As corollaries, some known results about supersoluble groups, classes of groups with $𝔉$-subnormal cyclic primary and Sylow subgroups are obtained. Also the new characterization of the class of groups whose extreme subgroups all belong $𝔉$, where $𝔉$ is a hereditary saturated formation, is obtained.

Let $G$ be a finite group and $H$ a subgroup of $G$. Then $H$ is said to be $s$-permutably embedded in $G$ if a Sylow $p$-subgroup of $H$ is also a Sylow $p$-subgroup of some $s$-permutable subgroup of $G$ for every prime dividing the order of $H$. We say that $H$ is nearly$S\mathrm{\Phi }$-embedded in $G$ if $G$ has a normal subgroup $N$ such that $HN$ is $s$-permutable in $G$ and $H\cap N\le \mathrm{\Phi }(H)H_{seG}$, where $H_{seG}$ is the subgroup of $H$ generated by all those subgroups of $H$ which are $s$-permutably embedded in $G$. In this paper, we study the properties of the nearly $S\mathrm{\Phi }$-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.

In this survey we show the influence of the abnormal maximal subgroups of finite groups in their structure.

If a subgroup H of a finite group G covers or avoids every chief factor of G, then we say that H has the cover and avoidance property in G or H is a CAP-subgroup of G. If H just covers or avoids the chief factors of a given chief series of G, then H is called partial CAP-subgroup of G. These subgroup embedding properties are extremely useful to clear up the group structure. The main aim of this survey is to present some recent results about the cover and avoidance properties in the context of the old ones.

In this paper we discuss some constructions and recent results of the theory of soluble and partially soluble finite groups.