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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2013, Volume 19, Number 3, Pages 316–320
(Mi timm991)
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On the $\mathfrak F$-residual of the direct product of finite groups
L. A. Shemetkov Francisk Skorina Gomel State University
Abstract:
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, formation, $\mathfrak F$-residual.
Received: 10.12.2012
Citation:
L. A. Shemetkov, “On the $\mathfrak F$-residual of the direct product of finite groups”, Trudy Inst. Mat. i Mekh. UrO RAN, 19, no. 3, 2013, 316–320
Linking options:
https://www.mathnet.ru/eng/timm991 https://www.mathnet.ru/eng/timm/v19/i3/p316
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Abstract page: | 297 | Full-text PDF : | 102 | References: | 68 | First page: | 5 |
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