On \(B\)-injectors of symmetric groups \(S_n\). (Q2915438)
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scientific article; zbMATH DE number 6083269
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On \(B\)-injectors of symmetric groups \(S_n\). |
scientific article; zbMATH DE number 6083269 |
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17 September 2012
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injectors
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symmetric groups
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maximal nilpotent subgroups
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Sylow subgroups
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On \(B\)-injectors of symmetric groups \(S_n\). (English)
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A \(B\)-injector of a finite group is a maximal nilpotent subgroup \(B\) of \(G\) satisfying \(d_2(G)=d_2(B)\) where \(d_2(X)=\max\{|A|:A\leq X\) and \(A\) is nilpotent of class at most \(2\}\).NEWLINENEWLINE The main result of this paper is the following theorem: Let \(\Omega\) be a finite set of size \(n\) and let \(B\leq S_\Omega\) be a \(B\)-injector of \(S_\Omega\). Then (a) if \(n\not\equiv 3\pmod 4\) then \(B\) is a Sylow \(2\)-subgroup of \(S_\Omega\); (b) if \(n\equiv 3\pmod 4\) then \(B=\langle d\rangle\times T\) where \(d\) is a \(3\)-cycle and \(T\) is a Sylow \(2\)-subgroup of \(C_{S_\Omega}(d)\).
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