A result about cosets (Q1904097)
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scientific article; zbMATH DE number 826765
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | A result about cosets |
scientific article; zbMATH DE number 826765 |
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A result about cosets (English)
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1 February 1996
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A finite group \(G\) is called in the paper a CSO-group if every proper nontrivial subgroup \(H\) of \(G\) has a coset \(Hx\) consisting of elements of equal order \(a(x, H)\). This being such a strong condition, the authors conjecture that every CSO-group must be soluble. They are able to make good use of the fact that a finite soluble group has normal subgroups of prime index in order to prove their main result: A soluble group \(G\) is a CSO-group if and only if \(G\) is a \(p\)-group and \(G\setminus \Phi(G)\) consists of elements of the same order.
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Frattini subgroup
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finite groups
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CSO-groups
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cosets
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elements of equal order
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finite soluble groups
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normal subgroups
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\(p\)-groups
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