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1. • CommentRowNumber1.
   • CommentAuthorjcmckeown
   • CommentTimeNov 26th 2012
   • PermaLink
   Author: jcmckeown
   Format: MarkdownItexNow there is [[Sylow p-subgroup]]. --- Is there a compilation, somewhere, of the results "the (obvious) automorphisms of a small $\mathfrak{A}$ $A$ are transitive on $A$'s maximal $\mathfrak{B}$s?" The only other example ready in my head is that the maximal tori in a compact Lie group are conjugate, but I know I've seen more.

   Now there is Sylow p-subgroup.

   ---

   Is there a compilation, somewhere, of the results “the (obvious) automorphisms of a small $\mathfrak{A}$ $A$ are transitive on $A$’s maximal $\mathfrak{B}$s?” The only other example ready in my head is that the maximal tori in a compact Lie group are conjugate, but I know I’ve seen more.

2. • CommentRowNumber2.
   • CommentAuthorTodd_Trimble
   • CommentTimeNov 26th 2012
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   Author: Todd_Trimble
   Format: MarkdownItexThe [[p-torsion]] article that is linked to from [[Sylow p-subgroup]] seems to refer only to abelian groups (?).

   The p-torsion article that is linked to from Sylow p-subgroup seems to refer only to abelian groups (?).

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeNov 26th 2012
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   Author: Urs
   Format: MarkdownItex> The p-torsion article I wasn't aware of that article. It overlaps a bit with the other article, _[[torsion subgroup]]_. I don't have time to merge them now, but I have added cross-links. (And I fixed the typo in the definition! :-)

   The p-torsion article

   I wasn’t aware of that article. It overlaps a bit with the other article, torsion subgroup. I don’t have time to merge them now, but I have added cross-links. (And I fixed the typo in the definition! :-)

4. • CommentRowNumber4.
   • CommentAuthorjcmckeown
   • CommentTimeNov 26th 2012
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   Author: jcmckeown
   Format: MarkdownItex@Todd, hmm... see, I was only looking for the proof of the unproved theorem now in Sylow, so I didn't look too closely... And now I see you're right, and therefore that the thing in the p-torsion page really should be called simply a *prime factor* of $G$, or at most a Smith submodule --- (er, did a Smith really write Smith's Algorithm?)

   @Todd, hmm… see, I was only looking for the proof of the unproved theorem now in Sylow, so I didn’t look too closely… And now I see you’re right, and therefore that the thing in the p-torsion page really should be called simply a prime factor of $G$, or at most a Smith submodule — (er, did a Smith really write Smith’s Algorithm?)

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 23rd 2018
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexAdded a proof of existence of Sylow subgroups mentioned by Benjamin Steinberg at the Caf&eacute;. <a href="https://ncatlab.org/nlab/revision/diff/Sylow+p-subgroup/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/Sylow+p-subgroup/7">v7</a>, <a href="https://ncatlab.org/nlab/show/Sylow+p-subgroup">current</a>

   Added a proof of existence of Sylow subgroups mentioned by Benjamin Steinberg at the Café.

   diff, v7, current

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 23rd 2018
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   Author: David_Corfield
   Format: MarkdownItexI'm probably being dim but shouldn't one add a condition to $H$. I mean, what's to stop $H$ being trivial. Or is that OK, and one can speak of the trivial group as a $p$-Sylow subgroup of itself for any $p$? Don't we need positive powers of $p$?

   I’m probably being dim but shouldn’t one add a condition to $H$. I mean, what’s to stop $H$ being trivial. Or is that OK, and one can speak of the trivial group as a $p$-Sylow subgroup of itself for any $p$? Don’t we need positive powers of $p$?

7. • CommentRowNumber7.
   • CommentAuthorTodd_Trimble
   • CommentTimeSep 23rd 2018
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   Author: Todd_Trimble
   Format: MarkdownItexNothing stops $H$ from being trivial or of order prime to $p$; there the $p$-Sylow subgroup is trivial as you surmise.

   Nothing stops $H$ from being trivial or of order prime to $p$; there the $p$-Sylow subgroup is trivial as you surmise.

8. • CommentRowNumber8.
   • CommentAuthorDavid_Corfield
   • CommentTimeSep 23rd 2018
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   Author: David_Corfield
   Format: MarkdownItexThere are some people who insist on $p$-groups and $p$-subgroups being nontrivial, e.g., Steven Roman in [Fundamentals of Group Theory](https://books.google.co.uk/books?id=eWkqG0aiVsMC), pp. 80-81, but I guess this is just a convention.

   There are some people who insist on $p$-groups and $p$-subgroups being nontrivial, e.g., Steven Roman in Fundamentals of Group Theory, pp. 80-81, but I guess this is just a convention.