Resumen de Transitivité pour une certaine "normalité faible" dans un groupe

C. Marchionna Tibiletti

• Let $H,K$ be subsets of a group $G$ and $Z = Z(G)\subset H\subseteq K\subseteq G$: then is defined "$H$ weak normal in $K$" if and only if $\forall k\in K\setminus Z, \exists h\in H\setminus Z, [h, k] = 1$. The subset $H$ of $G$ is defined "$Q$-complex" if and only if $\forall h\in H$ also $ \langle h\rangle\subset H$. The groups where the "weak normality" is transitive are characterized. The groups $G$ where the "weak normality" is transitive for the subsets $H\supset Z$, are those groups reducible (for permutability) where the distance (for permutability) is one. The groups where the "weak normality" is transitive for the $Q$-complex $K\supset Z$ are those groups where $\forall{x, y}\in G\setminus Z$ connexe (for permutability) $\exists n\in\mathbb{Z}, [x, y^n] = 1, y^n \notin Z(G)$. Other properties and a few examples will be also reported.