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.

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

]]>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 $p$?

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

]]>@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?)

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

]]>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.

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