Cyclic permutable subgroups of finite groups (Q2765548)
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scientific article; zbMATH DE number 1694843
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Cyclic permutable subgroups of finite groups |
scientific article; zbMATH DE number 1694843 |
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Cyclic permutable subgroups of finite groups (English)
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17 December 2002
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cyclic permutable subgroups
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finite groups
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groups of power automorphisms
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normal closures
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Abelian-by-cyclic groups
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A subgroup \(A\) of a group \(G\) is said to be permutable if \(AX=XA\) for all subgroups \(X\) of \(G\). Theorem: Let \(A\) be a cyclic permutable subgroup of odd order of a finite group \(G\). Then (1) \([A,G]\) is Abelian; (2) \(A\) acts by conjugation on \([A,G]\) as a group of power automorphisms; (3) the normal closure \(A^G\) of \(A\) in \(G\) is Abelian-by-cyclic.
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