This is very clear that if we have unique $p$-Sylow subgroup in a group G then it is normal in G by using second Sylow theorem, as single $p$-Sylow subgroup in a group is self conjugate to itself.... Now my ques is that suppose we have two distinct $p$-Sylow subgroups then why can we not use Sylow 2nd theorem here....why can't we use self conjugacy here???

  • $\begingroup$ Any two $p$-Sylow groups are conjugate. See, e.g., this $\endgroup$ – lulu Nov 9 '18 at 10:59
  • $\begingroup$ In you statement, do you mean exactly two distinct Sylow $p$-subgroups, or just an arbitrary pair? $\endgroup$ – Nicky Hekster Nov 9 '18 at 11:48
  • $\begingroup$ Just an arbitrary pair $\endgroup$ – Ibrahim Nov 9 '18 at 12:36
  • $\begingroup$ OK, then I concur with the answer of @freakish and +1 you both. $\endgroup$ – Nicky Hekster Nov 9 '18 at 13:21

The second Sylow theorem implies that every two Sylow $p$-subgroups are conjugates. So if there are two distinct Sylow $p$-subgroups then obviously none of them is normal, since normal subgroups don't have proper conjugates.

In other words: a Sylow $p$-subgroup is normal if and only if it is a unique Sylow $p$-subgroup.

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