$G$ is a group with $|G| = n$ and $p$ is the smallest prime dividing $|G|$ , then any subgroup of index $p$ is normal.
35602
If a subgroup has smallest prime index, then it is normal
13742
Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
122266
equivalent definition of normal subgroup
49122
Why commutator subgroup is normal to G?
58694
Which non-abelian finite groups have the property that every subgroup is normal?
97928
Normal subgroup of a normal subgroup
15706
"Every Group of order $216$ contains a normal subgroup of order $27$ or $9$
61064
Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.
138406
If N and every subgroup of N is normal in G then G/N is abelian .
55350
H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
133298
Assume $|G|=40$. Show that the subgroup of order $8$ is normal and unique.
38095
The Commutator Subgroup is Normal.
11731
Normal subgroups of $S_4$
13523
Are normal subgroups transitive?
65037
$H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
48273
If a maximal subgroup is normal, it has prime index