$H \leq G$ s.t. $|G : H|$ is the least prime dividing $|G|$. Show $H$ is normal in $G$.
7108
Normal subgroup of prime index
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
35602
If a subgroup has smallest prime index, then it is normal
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
54911
The preimage of a normal subgroup under a group homomorphism is normal
14379
Let $H \leq G$. If $G = H H^x$ then $G=H$.
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
154398
Show that H is a subgroup of G
59007
Suppose $|G| = 105$. Show that it is abelian if it has a normal $3$-Sylow subgroup.
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
7973
Groups of order $pq$ have a proper normal subgroup
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?