arXiv subjects
Publications and source records attributed to A. Martínez-Pastor.
The aim of this paper is to prove the following result: Let $π$ be a set of odd primes. If the group $G=AB$ is the product of two $π$-decomposable subgroups $A=A_π\times A_{π'}$ and $B=B_π\times B_{π'}$, then $G$ has a unique conjugacy class of Hall $π$-subgroups, and any $π$-subgroup is contained in a Hall $π$-subgroup (i.e. $G$ satisfies property $D_π$).
A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, $n$-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups $G = AB$ with subgroups $A,\ B$ such that $\langle a, b\rangle$ is soluble for all $a \in A$ and $b \in B$. In this case, the group $G$ is said to be an $\cal S$-connected product of the subgroups $A$ and $B$ for the class $\cal S$ of all finite soluble groups. Our main theorem states that $G = AB$ is $\cal S$-connected if and only if $[A,B]$ is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.
We provide structural criteria for some finite factorised groups $G = AB$ when the conjugacy class sizes in $G$ of certain $π$-elements in $A\cup B$ are either $π$-numbers or $π'$-numbers, for a set of primes $π$. In particular, we extend for products of groups some earlier results.
An element $g$ of a finite group $G$ is said to be vanishing in $G$ if there exists an irreducible character $χ$ of $G$ such that $χ(g)=0$; in this case, $g$ is also called a zero of $G$. The aim of this paper is to obtain structural properties of a factorised group $G=AB$ when we impose some conditions on prime power order elements $g\in A\cup B$ which are (non-)vanishing in $G$.
Let the group $G = AB$ be the product of the subgroups $A$ and $B$. We determine some structural properties of $G$ when the $p$-elements in $A\cup B$ have prime power indices in $G$, for some prime $p$. More generally, we also consider the case that all prime power order elements in $A\cup B$ have prime power indices in $G$. In particular, when $G = A = B$ we obtain as a consequence some known results.
We obtain some structural properties of a factorised group $G = AB$, given that the conjugacy class sizes of certain elements in $A\cup B$ are not divisible by $p^2$, for some prime $p$. The case when $G = AB$ is a mutually permutable product is especially considered.
Let $G$ be a finite group and $G_p$ be a Sylow $p$-subgroup of $G$ for a prime $p$ in $π(G)$, the set of all prime divisors of the order of $G$. The automiser $A_p(G)$ is defined to be the group $N_G(G_p)/G_pC_G(G_p)$. We define the Sylow graph $Γ_A(G)$ of the group $G$, with set of vertices $π(G)$, as follows: Two vertices $p,q\inπ(G)$ form an edge of $Γ_A(G)$ if either $q\inπ(A_p(G))$ or $p\in π(A_q(G))$. The following result is obtained: Theorem: Let $G$ be a finite almost simple group. Then the graph $Γ_A(G)$ is connected and has diameter at most 5. We also show how this result can be applied to derive information on the structure of a group from the normalizers of its Sylow subgroups.