Geodesic
On the existence of minimal $\tau$-closed totally saturated non-$\mathfrak H$-formations
Trudy Instituta matematiki, Tome 16 (2008) no. 1, pp. 67-72

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The article deals with finite groups. A $\tau$-closed totally saturated formation $\mathfrak F$ is called a minimal $\tau$-closed totally saturated non-$\mathfrak H$-formation (or an $\mathfrak H_\infty^\tau$-critical formation) if $\mathfrak F\not\subseteq\mathfrak H$, but all proper $\tau$-closed totally saturated subformations of $\mathfrak F$ are contained in $\mathfrak H$. Theorem. Let $\mathfrak F$ and $\mathfrak H$ be $\tau$-closed totally saturated formations, $\mathfrak F\not\subseteq\mathfrak H.$ Then $\mathfrak F$ has at least one minimal $\tau$-closed totally saturated non-$\mathfrak H$-formation. 
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     author = {V. G. Safonov},
     title = {On the existence of minimal $\tau$-closed totally saturated non-$\mathfrak H$-formations},
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V. G. Safonov. On the existence of minimal $\tau$-closed totally saturated non-$\mathfrak H$-formations. Trudy Instituta matematiki, Tome 16 (2008) no. 1, pp. 67-72. http://geodesic.mathdoc.fr/item/TIMB_2008_16_1_a11/