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???
Are any two distinct p-Sylow subgroups normal
finite-groupsgroup-theorysylow-theory
Best Answer
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.