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We characterize some classes of finite soluble groups. In particular, we prove that: a finite group $G$ is supersoluble if and only if $G$ has a normal subgroup $D$ such that $G/D$ is supersoluble and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every maximal subgroup $V$ of the generalized Fitting subgroup $F^{*}(G)$ of $G$; a finite soluble group $G$ is a $PST$-group (that is, Sylow permutability is a transitive relation on $G$) if and only if $G$ has a normal subgroup $D$ such that $G/D$ is nilpotent and $D$ avoids every chief factor of $G$ between $V^{G}$ and $V_{G}$ for every subnormal subgroup $A$ of $G$., Comment: 19 pages

In this paper, we discuss some well-known results and some open problems of the theory of $\sigma$-properties of a group related to the study of generalized $T$-groups., Comment: arXiv admin note: text overlap with arXiv:1709.06423, arXiv:1704.02509

Let ${\frak F}$ be a class of group and $G$ a finite group. Then a set $\Sigma $ of subgroups of $G$ is called a \emph{$G$-covering subgroup system} for the class ${\frak F}$ if $G\in {\frak F}$ whenever $\Sigma \subseteq {\frak F}$. We prove that: {\sl If a set of subgroups $\Sigma$ of $G$ contains at least one supplement to each maximal subgroup of every Sylow subgroup of $G$, then $\Sigma$ is a $G$-covering subgroup system for the classes of all $\sigma$-soluble and all $\sigma$-nilpotent groups, and for the class of all $\sigma$-soluble $P\sigma T$-groups.} This result gives positive answers to questions 19.87 and 19.88 from the Kourovka notebook., Comment: 8 pages

Let $G$ be a finite group and let $\mathfrak{F}$ be a hereditary saturated formation. We denote by $\mathbf{Z}_{\mathfrak{F}}(G)$ the product of all normal subgroups $N$ of $G$ such that every chief factor $H/K$ of $G$ below $N$ is $\mathfrak{F}$-central in $G$, that is, \[ (H/K) \rtimes (G/\mathbf{C}_{G}(H/K)) \in \mathfrak{F}. \]A subgroup $A \leq G$ is said to be $\mathfrak{F}$-subnormal in the sense of Kegel, or $K$-$\mathfrak{F}$-subnormal in $G$, if there is a subgroup chain \[ A = A_0 \leq A_1 \leq \ldots \leq A_n = G \] such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_i / (A_{i-1})_{A_i} \in \mathfrak{F}$ for all $i = 1, \ldots , n$. In this paper, we prove the following generalisation of Schenkman's Theorem on the centraliser of the nilpotent residual of a subnormal subgroup: Let $\mathfrak{F}$ be a hereditary saturated formation and let $S$ be a $K$-$\mathfrak{F}$-subnormal subgroup of $G$. If $\mathbf{Z}_{\mathfrak{F}}(E) = 1$ for every subgroup $E$ of $G$ such that $S \leq E$ then $\mathbf{C}_{G}(D) \leq D$, where $D = S^{\mathfrak{F}}$ is the $\mathfrak{F}$-residual of $S$., Comment: 9 pages

Let $G$ be a finite group and $\sigma$ a partition of the set of all? primes $\Bbb{P}$, that is, $\sigma =\{\sigma_i \mid i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_i$ and $\sigma_i\cap \sigma_j= \emptyset $ for all $i\ne j$. If $n$ is an integer, we write $\sigma(n)=\{\sigma_i \mid \sigma_{i}\cap \pi (n)\ne \emptyset \}$ and $\sigma (G)=\sigma (|G|)$. We call a graph $\Gamma$ with the set of all vertices $V(\Gamma)=\sigma (G)$ ($G\ne 1$) a $\sigma$-arithmetic graph of $G$, and we associate with $G\ne 1$ the following three directed $\sigma$-arithmetic graphs: (1) the $\sigma$-Hawkes graph $\Gamma_{H\sigma }(G)$ of $G$ is a $\sigma$-arithmetic graph of $G$ in which $(\sigma_i, \sigma_j)\in E(\Gamma_{H\sigma }(G))$ if $\sigma_j\in \sigma (G/F_{\{\sigma_i\}}(G))$; (2) the $\sigma$-Hall graph $\Gamma_{\sigma Hal}(G)$ of $G$ in which $(\sigma_i, \sigma_j)\in E(\Gamma_{\sigma Hal}(G))$ if for some Hall $\sigma_i$-subgroup $H$ of $G$ we have $\sigma_j\in \sigma (N_{G}(H)/HC_{G}(H))$; (3) the $\sigma$-Vasil'ev-Murashko graph $\Gamma_{{\mathfrak{N}_\sigma }}(G)$ of $G$ in which $(\sigma_i, \sigma_j)\in E(\Gamma_{{\mathfrak{N}_\sigma}}(G))$ if for some ${\mathfrak{N}_{\sigma }}$-critical subgroup $H$ of $G$ we have $\sigma_i \in \sigma (H)$ and $\sigma_j\in \sigma (H/F_{\{\sigma_i\}}(H))$. In this paper, we study the structure of $G$ depending on the properties of these three graphs of $G$.

Let $\mathfrak{F}$ be a class of finite groups and $G$ a finite group. Let ${\cal L}_{\mathfrak{F}}(G)$ be the set of all subgroups $A$ of $G$ with $A^{G}/A_{G}\in \mathfrak{F}$. A chief factor $H/K$ of $G$ is $\mathfrak{F}$-central in $G$ if $(H/K)\rtimes (G/C_{G}(H/K)) \in\mathfrak{F}$. We study the structure of $G$ under the hypothesis that every chief factor of $G$ between $A_{G}$ and $A^{G}$ is $\mathfrak{F}$-central in $G$ for every subgroup $A\in {\cal L}_{\mathfrak{F}}(G)$. As an application, we prove that a finite soluble group $G$ is a $PST$-group if and only if $A^{G}/A_{G}\leq Z_{\infty}(G/A_{G})$ for every subgroup $A\in {\cal L}_{\mathfrak{N}}(G)$, where $\mathfrak{N}$ is the class of all nilpotent groups.

Throughout this paper, all groups are finite. Let $\sigma =\{\sigma_{i} | i\in I \}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, the symbol $\sigma (n)$ denotes the set $\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$. The integers $n$ and $m$ are called $\sigma$-coprime if $\sigma (n)\cap \sigma (m)=\emptyset$. Let $t > 1$ be a natural number and let $\mathfrak{F}$ be a class of groups. Then we say that $\mathfrak{F}$ is $\Sigma_{t}^{\sigma}$-closed provided $\mathfrak{F}$ contains each group $G$ with subgroups $A_{1}, \ldots , A_{t}\in \mathfrak{F}$ whose indices $|G:A_{1}|$, $\ldots$, $|G:A_{t}|$ are pairwise $\sigma$-coprime. In this paper, we study $\Sigma_{t}^{\sigma}$-closed classes of finite groups., Comment: 12 pages

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set $\Bbb{P}$ of all primes and $G$ a finite group. A chief factor $H/K$ of $G$ is said to be $\sigma$-central if the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is a $\sigma_{i}$-group for some $i=i(H/K)$. $G$ is called $\sigma$-nilpotent if every chief factor of $G$ is $\sigma$-central. We say that $G$ is semi-${\sigma}$-nilpotent (respectively weakly semi-${\sigma}$-nilpotent) if the normalizer $N_{G}(A)$ of every non-normal (respectively every non-subnormal) $\sigma$-nilpotent subgroup $A$ of $G$ is $\sigma$-nilpotent. In this paper we determine the structure of finite semi-${\sigma}$-nilpotent and weakly semi-${\sigma}$-nilpotent groups., Comment: 16 pages

Let $G$ be a finite group and $\sigma =\{\sigma_{i} | i\in I\}$ some partition of the set of all primes $\Bbb{P}$, that is, $\sigma =\{\sigma_{i} | i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} \sigma_{i}$ and $\sigma_{i}\cap \sigma_{j}= \emptyset $ for all $i\ne j$. We say that $G$ is $\sigma$-primary if $G$ is a $\sigma _{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${\sigma}$-subnormal in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $\sigma$-primary for all $i=1, \ldots, n$, modular in $G$ if the following conditions hold: (i) $\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (ii) $\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $A \leq Z$. In this paper, a subgroup $A$ of $G$ is called $\sigma$-quasinormal in $G$ if $L$ is modular and ${\sigma}$-subnormal in $G$. We study $\sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $\sigma$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is $\sigma$-primary., Comment: 9 pages

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $\sigma (G)=\{\sigma _{i} : \sigma _{i}\cap \pi (G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i$ such that $\sigma _{i}\in \sigma (G)$. We say that $G$ is $\sigma$-full if $G$ possesses a complete Hall $\sigma $-set. A complete Hall $\sigma $-set $\cal H$ of $G$ is said to be a $\sigma$-basis of $G$ if every two subgroups $A, B \in\cal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $\sigma$-basis. In particular, we prove that if $G$ has a a $\sigma$-basis, then $G$ is generalized $\sigma$-soluble, that is, $G$ has a complete Hall $\sigma $-set and for every chief factor $H/K$ of $G$ we have $|\sigma (H/K)|\leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $\sigma$-full. Then every complete Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$ if and only if $G$ is generalized $\sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $\sigma (H/K)=\sigma (G/C_{G}(H/K))$ or $\sigma (H/K) =\{\sigma _{i}\}$ and $G/C_{G}(H/K)$ is a $\sigma _{i} \cup \sigma _{j}$-group for some $i\ne j$., Comment: 13 pages

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and let $G$ be a finite group. Then $G$ is said to be $\sigma $-full if $G$ has a Hall $\sigma _{i}$-subgroup for all $i$. A subgroup $A$ of $G$ is said to be $\sigma$-permutable in $G$ provided $G$ is $\sigma $-full and $A$ permutes with all Hall $\sigma _{i}$-subgroups $H$ of $G$ (that is, $AH=HA$) for all $i$. We obtain a characterization of finite groups $G$ in which $\sigma$-permutability is a transitive relation in $G$, that is, if $K$ is a ${\sigma}$-permutable subgroup of $H$ and $H$ is a ${\sigma}$-permutable subgroup of $G$, then $K$ is a ${\sigma}$-permutable subgroup of $G$., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1704.02509

Let $\frak {F}$ be a class of group. A subgroup $A$ of a finite group $G$ is said to be $K$-$\mathfrak{F}$-subnormal in $G$ if there is a subgroup chain $$A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$$ such that either $A_{i-1} \trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}} \in \mathfrak{F}$ for all $i=1, \ldots , n$. A formation $\frak {F}$ is said to be $K$-lattice provided in every finite group $G$ the set of all its $K$-$\mathfrak{F}$-subnormal subgroups forms a sublattice of the lattice of all subgroups of $G$. In this paper we consider some new applications of the theory of $K$-lattice formations. In particular, we prove the following Theorem A. Let $\mathfrak{F}$ be a hereditary $K$-lattice saturated formation containing all nilpotent groups. (i) If every $\mathfrak{F}$-critical subgroup $H$ of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$ with $H/F(H)\in {\mathfrak{F}}$, then $G/F(G)\in {\mathfrak{F}}$. (ii) If every Schmidt subgroup of $G$ is $K$-$\mathfrak{F}$-subnormal in $G$, then $G/G_{\mathfrak{F}}$ is abelian., Comment: 11 pages

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a $\sigma _{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $\sigma _{i}\in \sigma $ and ${\cal H}$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i \in I$ such that $\sigma _{i}\cap \pi (G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be ${\sigma}$-permutable in $G$ if $G$ has a complete Hall $\sigma$-set $\cal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \cal H$. We obtain characterizations of finite $\sigma$-soluble groups $G$ in which $\sigma$-permutability is a transitive relation in $G$., Comment: 15 pages. arXiv admin note: text overlap with arXiv:1611.06569

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $\sigma (G) =\{\sigma_{i} |\sigma_{i}\cap \pi (G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\cal H}$ contains exactly one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in \sigma (G)$. A subgroup $A$ of $G$ is said to be ${\sigma}$-permutable in $G$ if $G$ possesses a complete Hall $\sigma $-set and $A$ permutes with each Hall $\sigma_{i}$-subgroup $H$ of $G$, that is, $AH=HA$ for all $i \in I$. We characterize finite groups with distributive lattice of the ${\sigma}$-permutable subgroups., Comment: 13 pages

Let $G$ be a finite group. Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $n$ an integer. We write $\sigma (n) =\{\sigma_{i} |\sigma_{i}\cap \pi (n)\ne \emptyset \}$, $\sigma (G) =\sigma (|G|)$. A set $ {\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma $-set of $G$ if every member of ${\cal H}\setminus \{1\}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}$ and ${\cal H}$ contains exact one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in \sigma (G)$. A subgroup $A$ of $G$ is called: (i) a $\sigma$-Hall subgroup of $G$ if $\sigma (|A|) \cap \sigma (|G:A|)=\emptyset$; (ii) ${\sigma}$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set ${\cal H}$ such that $AH^{x}=H^{x}A$ for all $H\in {\cal H}$ and all $x\in G$. We say that a subgroup $A$ of $G$ is $H_{\sigma}$-permutably embedded in $G$ if $A$ is a ${\sigma}$-Hall subgroup of some ${\sigma}$-permutable subgroup of $G$. We study finite groups $G$ having an $H_{\sigma}$-permutably embedded subgroup of order $|A|$ for each subgroup $A$ of $G$. Some known results are generalized., Comment: 15 pages

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. $G$ is said to be \emph{$\sigma$-soluble} if every chief factor $H/K$ of $G$ is a $\sigma_{i}$-group for some $i=i(H/K)$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\cal H}$ contains exact one Hall $\sigma_{i}$-subgroup of $G$ for every $i \in I$ such that $\sigma_{i}\cap \pi (G)\ne \emptyset$. A subgroup $A$ of $G$ is said to be \emph{${\sigma}$-permutable} or \emph{${\sigma}$-quasinormal} in $G$ if $G$ has a complete Hall $\sigma$-set $\cal H$ such that $AH^{x}=H^{x}A$ for all $x\in G$ and all $H\in \cal H$. We obtain a characterization of finite $\sigma$-soluble groups $G$ in which $\sigma$-quasinormality is a transitive relation in $G$., Comment: 11 pages

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$, $G$ a finite group and $\sigma (G) =\{\sigma_{i} |\sigma_{i}\cap \pi (G)\ne \emptyset \}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$ for some $\sigma_{i}\in \sigma $ and ${\cal H}$ contains exact one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in \sigma (G)$. A subgroup $H$ of $G$ is said to be: \emph{${\sigma}$-semipermutable in $G$ with respect to ${\cal H}$} if $HH_{i}^{x}=H_{i}^{x}H$ for all $x\in G$ and all $H_i\in {\cal H}$ such that $(|H|, |H_{i}|)=1$; \emph{${\sigma}$-semipermutable in $G$} if $H$ is ${\sigma}$-semipermutable in $G$ with respect to some complete Hall $\sigma $-set of $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are ${\sigma}$-semipermutable in $G$., Comment: 14 pages

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. A set ${\cal H}$ of subgroups of $G$ is said to be a \emph{complete Hall $\sigma $-set} of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma_{i}$-subgroup of $G$, for some $i\in I$, and $\cal H$ contains exact one Hall $\sigma_{i}$-subgroup of $G$ for every $\sigma_{i}\in \sigma (G)$. A subgroup $H$ of $G$ is said to be: \emph{$\sigma$-permutable} or \emph{$\sigma$-quasinormal} in $G$ if $G$ possesses a complete Hall $\sigma$-set set ${\cal H}$ such that $HA^{x}=A^{x}H$ for all $A\in {\cal H}$ and $x\in G$: \emph{${\sigma}$-subnormal} in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{t}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is a finite $\sigma_{i}$-group for some $\sigma_{i}\in \sigma$ for all $i=1, \ldots t$. If each $n$-maximal subgroup of $G$ is $\sigma$-subnormal ($\sigma$-quasinormal, respectively) in $G$ but, in the case $ n > 1$, some $(n-1)$-maximal subgroup is not $\sigma$-subnormal (not $\sigma$-quasinormal, respectively)) in $G$, we write $m_{\sigma}(G)=n$ ($m_{\sigma q}(G)=n$, respectively). In this paper, we show that the parameters $m_{\sigma}(G)$ and $m_{\sigma q}(G)$ make possible to bound the $\sigma$-nilpotent length $ \ l_{\sigma}(G)$ (see below the definitions of the terms employed), the rank $r(G)$ and the number $|\pi (G)|$ of all distinct primes dividing the order $|G|$ of a finite soluble group $G$. We also give conditions under which a finite group is $\sigma$-soluble or $\sigma$-nilpotent, and describe the structure of a finite soluble group $G$ in the case when $m_{\sigma}(G)=|\pi (G)|$. Some known results are generalized.

A proper subgroup $H$ of a group $G$ is said to be: $\Bbb{P}$-subnormal in $G$ if there exists a chain of subgroups $H=H_0 < H_1< ... < H_{n}=G$ such that $|H_{i}:H_{i-1}|$ is a prime for $i=1,...,n$; $\Bbb{P}$-abnormal in $G$ if for every two subgroups $K\leq L$ of $G$, where $H\leq K$, $|L:K|$ is not a prime. In this paper we describe finite groups in which every non-identity subgroup is either $\Bbb{P}$-subnormal or $\Bbb{P}$-abnormal.

We consider some applications of the theory of generalized Ore supplement conditions in the study of finite groups.

Let $H$ and $B$ be subgroups of a finite group $G$ such that $G=N_{G}(H)B$. Then we say that $H$ is \emph{quasipermutable} (respectively \emph{$S$-quasipermutable}) in $G$ provided $H$ permutes with $B$ and with every subgroup (respectively with every Sylow subgroup) $A$ of $B$ such that $(|H|, |A|)=1$. In this paper we analyze the influence of $S$-quasipermutable and quasipermutable subgroups on the structure of $G$. As an application, we give new characterizations of soluble $PST$-groups.

We describe finite soluble groups in which every $n$-maximal subgroup is $\mathfrak F$-subnormal.

Let $\cal F$ be a class of groups. A chief factor $H/K$ of a group $G$ is called \emph{${\cal F}$-central in $G$} provided $(H/K)\rtimes (G/C_{G}(H/K)) \in {\cal F}$. We write $Z_{\pi{\cal F}}(G)$ to denote the product of all normal subgroups of $G$ whose $G$-chief factors of order divisible by at least one prime in $\pi$ are $\cal F$-central. We call $Z_{\pi{\cal F}}(G)$ the $\pi{\cal F}$-hypercentre of $G$. A subgroup $U$ of a group $G$ is called \emph{$\cal F$-maximal} in $G$ provided that (a) $U\in {\cal F}$, and (b) if $U\leq V\leq G$ and $V\in {\cal F}$, then $U=V$. In this paper we study the properties of the intersection of all $\cal F$-maximal subgroups of a finite group. In particular, we analyze the condition under which $Z_{\pi{\cal F}}(G)$ coincides with the intersection of all $\cal F$-maximal subgroups of $G$., Comment: arXiv admin note: text overlap with arXiv:1103.4729

In the paper new criteria of existence and conjugacy of Hall subgroups of finite groups are given.

We investigate the influence of the intersection of the $\cal F$-maximal subgroups on the structure of a finite group. In particular, answering a question of L.A Shemetkov we give conditions under which a hereditary saturated formation $\cal F$ has a property that for any finite group $G$, the ${\cal F}$-hypercentre of $G$ coincides with the intersection of all $\cal F$-maximal subgroups of $G$.