Related papers: On the intersections of solvable Hall subgroups in…

Let $\sigma =\{\sigma_{i} | i\in I\}$ be some partition of the set of all primes $\Bbb{P}$ and $\Pi$ a non-empty subset of the set $\sigma$. A set ${\cal H}$ of subgroups of a finite group $G$ is said to be a \emph{ complete Hall $\Pi…

Let $G$ be a finite group and $H$ a core-free subgroup of $G$. We will show that if there exists a solvable, generating transversal of $H$ in $G$, then $G$ is a solvable group. Further, if $S$ is a generating transversal of $H$ in $G$ and…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the partial $ \Pi $-property in $ G $ if there exists a $G$-chief series $ \varGamma_{G}: 1 =G_{0} < G_{1} < \cdot\cdot\cdot < G_{n}= G $ of $ G $ such that $ | G…

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…

Let $ H $ be a subgroup of a finite group $ G $. We say that $ H $ satisfies the $ \Pi $-property in $ G $ if $ | G/K : N_{G/K}((H \cap L)K/K)| $ is a $ \pi(( H \cap L)K/K ) $-number for any chief factor $ L/K $ of $ G $; and we call that $…

In this paper, we show that each finite group $G$ containing at most $p^2$ Sylow $p$-subgroups for each odd prime number $p$, is a solvable group. In fact, we give a positive answer to the conjecture in \cite{Rob}.

Let $A$ and $G$ be finite groups such that $A$ acts coprimely on $G$ by automorphisms, we first prove some results on the solvability of finite groups in which some maximal $A$-invariant subgroups have indices a prime or the square of a…

Let $G$ be a finite group and $A$ be a subgroup of $G$. Then $A$ is called a $p$-$CAP$-subgroup of $G$, if $A$ covers or avoids every $pd$-chief factor of $G$. A subgroup $H$ of $G$ is said to be an $ICPC$-subgroup of $G$, if $H \cap [H,G]…

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…

The solubilizer of an element $x$ of a profinite group $G$ is the set of the elements $y$ of $G$ such that the subgroup of $G$ generated by $x$ and $y$ is prosoluble. We propose the following conjecture: the solubilizer of $x$ in $G$ has…

Let $G$ be a finite group, $\pi$ be a set of primes, and define $H_{\pi}(G)$ to be the subgroup generated by all elements of $G$ which do not have prime order for every prime in $\pi$. In this paper, we investigate some basic properties of…

An integral of a group $G$ is a group $H$ whose commutator subgroup is isomorphic to $G$. In this paper, we prove that the integrability of a finite group is a decidable problem.

A group $G$ is integrable if it is isomorphic to the derived subgroup of a group $H$; that is, if $H'\simeq G$, and in this case $H$ is an integral of $G$. If $G$ is a subgroup of $U$, we say that $G$ is integrable within $U$ if $G=H'$ for…

We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups which can easily be verified from the character table.

Let $G\in\{p,q\}^*$ be a finite group with trivial center, where $p,q\in\pi(G)$ and $p>q>5$. In the present paper it is proved that $|G|_{\{p,q\}}=|G||_{\{p,q\}}$; in particular $C_G(g)\cap C_G(h)=1$ for every $p$-element $g$ and every…

A $p$-subgroup $H$ of a finite group $G$ is said to satisfy partial $S$-$\Pi$-property in $G$ if $G$ has a chief series $\Gamma_{G}: 1=G_{0}<G_{1}<\cdots<G_{n}=G$ such that for every $G$-chief factor $G_{i}/G_{i-1}$ $(1\leqslant i\leqslant…

The aim of this paper is to prove the following result: Let $\pi$ be a set of odd primes. If the group $G=AB$ is the product of two $\pi$-decomposable subgroups $A=A_\pi \times A_{\pi'}$ and $B=B_\pi \times B_{\pi'}$, then $G$ has a unique…

Let $\pi$ be a set of primes. By H.Wielandt definition, {\it Sylow $\pi$-theorem} holds for a finite group $G$ if all maximal $\pi$-subgroups of $G$ are conjugate. In the paper, the following statement is proven. Assume that $\pi$ is a…

A new conjecture on characters of finite groups, related to the McKay conjecture, was proposed recently by the first and third authors. In this paper, we prove it for $p$-solvable groups when $p$ is odd.

Suppose that $x$, $y$ are elements of a finite group $G$ lying in conjugacy classes of coprime sizes. We prove that $\langle x^G \rangle \cap \langle y^G \rangle$ is an abelian normal subgroup of $G$ and, as a consequence, that if $x$ and…