Geodesic
On the $\mathfrak F$-residual of the direct product of finite groups
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 316-320

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Let $\pi$ be a subset of the set $\mathbb P$ of all primes, and let $\pi'=\mathbb P\backslash\pi$. A formation $\mathfrak F$ is called $\pi'$-saturated if $G/O_{\pi'}(\Phi(G))\in\mathfrak F$ implies $G\in\mathfrak F$. If $\mathfrak F$ is a nonempty $\pi'$-saturated formation of $\pi$-soluble groups, then it is proved that $(A\otimes B)^\mathfrak F=A^\mathfrak F\otimes B^\mathfrak F$ for any finite groups $A$ and $B$. In the case $\pi=\mathbb P$, this result was proved by K. Doerk and T. Hawkes in 1978.
Keywords: finite group, direct product, $\mathfrak F$-residual.
Mots-clés : formation 
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L. A. Shemetkov. On the $\mathfrak F$-residual of the direct product of finite groups. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 3, pp. 316-320. http://geodesic.mathdoc.fr/item/TIMM_2013_19_3_a33/