On saturated formations which are special relative to the strong covering-avoidance property

Abstract:

Let $\mathfrak {F}$ be a saturated formation of finite soluble groups. Let $G$ be a finite soluble group and $F$ an $\mathfrak {F}$-projector of $G$. Then $F$ is said to satisfy the strong covering-avoidance property if (i) $F$ either covers or avoids each chief factor of $G$, anaaa$F \cap L/F \cap K$ is a chief factor of $F$ whenever $L/K$ is a chief factor of $G$ coverzd by $F$Let${\mathcal {C}_\mathfrak {F}}$ denote the class of all finite soluble $G$ in which the $\mathfrak {F}$-projectors satisfy the strong covering-avoidance property. ${\mathcal {C}_\mathfrak {F}}$ is a formation. Let ${\mathcal {Y}_\mathfrak {F}}$ be the class of groups $G$ in which an $\mathfrak {F}$-normalizer is also an $\mathfrak {F}$-projector. ${\mathcal {Y}_\mathfrak {F}}$aa is a formation studied by Klaus Doerk. Note that ${\mathcal {Y}_\mathfrak {F}} \subseteq {\mathcal {C}_\mathfrak {F}}$. $\mathfrak {F}$ is said to be $\mathcal {C}$-special if ${\mathcal {C}_\mathfrak {F}} = {\mathcal {Y}_\mathfrak {F}}$. he puraaaaaaaaaaas note is to sdy $\mathcal {C}$-special formations. Two characterizations of $\mathcal {C}$-special formations are given. Let $i$ be a positive integer and let ${\mathfrak {N}^{(i)}}$ denote the class of finite soluble groups $G$ whose Fitting length is at most $i$. Then ${\mathfrak {N}^{(i)}}$ is $\mathcal {C}$-special. Finally, the formation ${\mathcal {C}_\mathfrak {F}}$aaaaaturated if and only if $\mathfrak {F}$ is the class of all finite soluble groups.