A sufficient condition of \(\pi'\)-closed groups (Q2704563)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A sufficient condition of \(\pi'\)-closed groups |
scientific article |
Statements
22 July 2001
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\(\pi\)-closed groups
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\(\pi\)-homogeneous groups
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minimal simple groups
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normal Hall subgroups
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finite groups
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0.92700565
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0.9021637
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0.88133126
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A sufficient condition of \(\pi'\)-closed groups (English)
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Let \(G\) be a finite group, \(\pi(G)\) the set of prime divisors of \(|G|\) and \(\pi\subseteq\pi(G)\). \(G\) is called a \(\pi\)-homogeneous group if \(N_G(H)/C_G(H)\) is a \(\pi\)-group for every \(\pi\)-subgroup \(H\) of \(G\), and \(G\) is called a \(\pi'\)-closed group if \(G\) has a normal Hall \(\pi'\)-subgroup. Using the classification of the minimal simple groups, the author shows that the minimal simple groups do not contain any element of order 10. Then he proves that if every nonabelian simple section of \(G\) contains elements of order 10, then \(G\) is \(\pi'\)-closed if and only if \(G\) is \(\pi\)-homogeneous.
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