Related papers: On the intersections of Sylow subgroups in almost …

Let $G$ be a finite group and let $H_p$ be a Sylow $p$-subgroup of $G$. A recent conjecture of Lisi and Sabatini asserts the existence of an element $x \in G$ such that $H_p \cap H_p^x$ is inclusion-minimal in the set $\{H_p \cap H_p^g…

Let $H$ be a subgroup of a finite group $G$. We say that $H$ satisfies partial $\Pi$-property in $G$ if there exists a chief series $\mathit{\Gamma}_G:1=G_0<G_1<\cdots<G_n=G$ of $G$ such that for every $G$-chief factor $G_i/G_{i-1}$ ($1\leq…

Denote by $\nu_p(G)$ the number of Sylow $p$-subgroups of $G$. It is not difficult to see that $\nu_p(H)\leq\nu_p(G)$ for $H\leq G$, however $\nu_p(H)$ does not divide $\nu_p(G)$ in general. In this paper we reduce the question whether…

For a prime $p$, a $p$-subgroup of a finite group $G$ is said to be large if and only if $Q= F^*(N_G(Q))$ and, for all $1 \neq U \le Z(Q)$, $N_G(U) \le N_G(Q)$. In this article we determine those groups $G$ which have a large subgroup and…

Let $G$ be an almost simple sporadic group and let $H$ be a soluble subgroup of $G$. In this paper we prove that there exists $x,y \in G$ such that $H \cap H^x \cap H^y=1$, which is equivalent to the bound $b(G,H) \leqslant 3$ with respect…

Suppose $p$ is a prime and $S$ is a Sylow $p$-subgroup of a finite group $G$. If $S$ is normal in $G$, then $Z(S)$ is the direct product of $S \cap Z(G)$ with $[Z(S), G]$. We prove an analogous result for all groups except in some cases…

Let $G$ be a finite group and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. We conjecture that there exists $x \in G$ such that $P_i \cap P_i^x$ is inclusion-minimal in $\{ P_i \cap P_i^g : g \in G\}$ for all…

Let $G$ be a finite group and $n_p(G)$ the number of Sylow $p$-subgroups of $G$. In this paper, we prove if $n_p(G)<p^2$ then almost all numbers $n_p(G)$ are a power of a prime.

In the paper we consider the following conjecture: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. The…

Let $G$ be a finite simple group of Lie type and let $P$ be a Sylow $2$-subgroup of $G$. In this paper, we prove that for any nontrivial element $x \in G$, there exists $g \in G$ such that $G = \langle P, x^g \rangle$. By combining this…

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…

Given two subsets $X,Y$ of a finite group $G$, we write $\Pr(X,Y)$ for the probability that random elements $x \in X$ and $y \in Y$ commute. If $X,Y$ are subgroups, we denote by $\Pr^*(X,Y)$ the maximum real number $\epsilon$ with the…

Let $p$ be a prime number, $G$ be a $p$-solvable finite group and $P$ be a Sylow $p$-subgroup of $G$. We prove that $G$ is $p$-supersolvable if $N_G(P)$ is $p$-supersolvable and if there is a subgroup $H$ of $P$ with $P' \le H \le \Phi(P)$…

We prove a natural generalization of Szep's conjecture. Given an almost simple group $G$ with socle not isomorphic to an orthogonal group having Witt defect zero, we classify all possible group elements $x,y\in G\setminus\{1\}$ with $G={\bf…

It is proved that a profinite group $G$ has fewer than $2^{\aleph_0}$ conjugacy classes of $p$-elements for an odd prime $p$ if and only if its $p$-Sylow subgroups are finite. (Here, by a $p$-element one understands an element that either…

For G = PSL(2,p^f) denote by ZG the integral group ring, by V(ZG) the group of normalized units of ZG and let r be a prime different from p. Using the so called HeLP-method we prove, that units of r-power order in V(ZG) are rationally…

It is shown that for a solvable subgroup $G$ of an almost simple group $S$ which socle is isomorphic to $A_n$ $ (n\ge5)$ there are $x,y,z,t \in S$ such that $G \cap G^x \cap G^y \cap G^z \cap G^t =1.$

In 2001 Liebeck and Pyber showed that a finite simple group of Lie type is a product of $ 25 $ carefully chosen unipotent Sylow subgroups. Later, in a series of works it was shown that $ 4 $ unipotent Sylow subgroups suffice. We prove that…

The Gruenberg-Kegel graph $\Gamma(G)$ of a finite group $G$ is the graph whose vertex set is the set of prime divisors of $|G|$ and in which two distinct vertices $r$ and $s$ are adjacent if and only if there exists an element of order $rs$…

In this short article, we give a summary of the Sylow $p$-subgroups of the finite simple groups of classical Lie type.