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$|G|=p^{a}m$

the conjugacy part of sylow's theorems , what does this denote? the first theorem ( the exist of sylow $p$-subgroup ) ? or the second one ( every $p$-subgroup is contained in some conjugate of sylow $p$-subgroup ) ? or the third one ( the number of sylpw $p$-subgroups $n_p$ is on the form $1 + kp$ for some positive integar k and that $n_p$ divides $ m$ ) ?

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    $\begingroup$ Every Sylow $p$-subgroup is conjugate to every other Sylow $p$-subgroup. Thus, if there is only one Sylow $p$-subgroup, it must be normal. $\endgroup$ – Clayton Mar 13 '13 at 4:54
  • $\begingroup$ The statement that ''Sylow-$p$ subgroups are conjugate" is clearly very stronger statement; it says the Sylow-$p$ subgroups are not only ''isomorphic'', but the isomorphism is induced by conjugation within $G$ itself. $\endgroup$ – Beginner Mar 15 '13 at 3:21
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That'd be the second one, as it is the one that contains the word "conjugate," or perhaps its corollary that all Sylow $p$-subgroups are (pairwise) conjugate.

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