# Are any two distinct p-Sylow subgroups normal?

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???

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