If a Sylow p-subgroup is a normal subgroup of G, it must be the only one, i.e. np = 1. And vice versa, if np = 1, then the one Sylow p-subgroup is a normal subgroup of G. ... In particular, if G is abelian, any subgroup is normal. So abelian groups have exactly one Sylow p-subgroup for each p.

Are all sylow P-subgroups normal?

If G has precisely one Sylow p-subgroup, it must be normal from Unique Subgroup of a Given Order is Normal. Suppose a Sylow p-subgroup P is normal. Then it equals its conjugates. Thus, by the Third Sylow Theorem, there can be only one such Sylow p-subgroup.

What is a normal Sylow subgroup?

A subgroup of a finite group is termed a normal Sylow subgroup if it satisfies the following equivalent conditions: It is a Sylow subgroup, and is normal in the whole group. It is a Sylow subgroup, and is subnormal in the whole group. It is a Sylow subgroup, and is characteristic in the whole group.