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Revision difference : \v{C}unihin's theorem |
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Version 6 |
| \begin{thm}[\v{C}unihin] |
Let $G$ be a finite, \PMlinkname{$\pi$-separable group}{Seperable}, for some set $\pi$ of primes. Then if $H$ is a maximal \PMlinkname{$\pi$-subgroup}{PiGroupsAndPiGroups} of $G$, the index of $H$ in $G$, $|G:H|$, is coprime to all elements of $\pi$ and all such subgroups are conjugate. Such a subgroup is called a Hall $\pi$-subgroup. For $\pi=\{p\}$, this essentially reduces to the Sylow theorems (with unnecessary hypotheses). |
| Let $G$ be a finite, \PMlinkname{$\pi$-separable group}{Seperable}, for some set $\pi$ of primes. Then |
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| \begin{itemize} |
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| any \PMlinkname{$\pi$-subgroup}{PiGroupsAndPiGroups} is contained in a \PMlinkname{Hall $\pi$-subgroup}{HallPiSubgroup}, and |
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| any two Hall $\pi$-subgroups are conjugate of one another |
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| \end{itemize} |
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| \end{thm} |
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| \textbf{Remarks} |
If $G$ is solvable, it is $\pi$-separable for all $\pi$, so such subgroups exist for all $\pi$. This result is often called Hall's theorem. |
| \begin{enumerate} |
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| For $\pi=\{p\}$, this essentially reduces to the Sylow theorems (with unnecessary hypotheses). |
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| If $G$ is solvable, it is $\pi$-separable for all $\pi$, so such subgroups exist for all $\pi$. This result is often called \PMlinkname{Hall's theorem}{HallsTheorem}. |
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| \end{enumerate} |
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| \begin{thebibliography}{9} |
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| \bibitem{dr} Derek J.S. Robinson. \emph{A Course in the Theory of Groups}, second edition. Springer (1995) |
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| \end{thebibliography} |
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