On direct products of classes of finite groups

N. N. Vorob'ev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 3, pp. 67-74

Résumé

All groups considered are finite. A set $\{\mathfrak F_i\mid i\in I\}$ of non-empty classes of groups $\mathfrak F_i$ is called orthogonal (Skiba, 1999) if: (1) either $|I|=1$ or $|I|>1$ and (2) $\mathfrak F_i\cap\mathfrak F_j=(1)$ for all $i,j\in I$, $i\ne j$. For any orthogonal system of classes $\{\mathfrak F_i\mid i\in I\}$ we denote by $\bigotimes_{i\in I}\mathfrak F_i$ the set of all groups isomorphic to groups of the form $A_1\times A_2\times\dots\times A_t$, where $A_1\in\mathfrak F_{i_1}$, $A_2\in\mathfrak F_{i_2}$, $\dots$, $A_t\in\mathfrak F_{i_t}$ for some $i_1,i_2,\dots,i_t\in I$. Let $\mathfrak F$ be a non-empty class of groups. The class $\mathfrak F$ is said to be the direct product of classes $\{\mathfrak F_i\mid i\in I\}$ if the set $\{\mathfrak F_i\mid i\in I\}$ is an orthogonal system of classes and $\bigotimes_{i\in I}\mathfrak F_i$. Let $\mathfrak F=\bigotimes_{i\in I}\mathfrak F_i$, where $\mathfrak F_i$ is a Fitting class. We prove that the Fitting class $\mathfrak F$ is $n$-multiply $\omega$-local if and only if each of the Fitting classes $\mathfrak F_i$ is $n$-multiply $\omega$-local.

Détail

Keywords: finite group, Fitting class, $n$-multiply $\omega$-local Fitting class.