Proving that for $|G|= p^n$, that some subgroup $N$ is normal in G, where $|N|=p^{n-1}$
7108
Normal subgroup of prime index
78580
Is $SL_n(\mathbb{R})$ a normal subgroup in $GL_n(\mathbb{R})$?
103423
Normal subgroup where G has order prime
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
19720
A normal subgroup intersects the center of the $p$-group nontrivially
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
64269
Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
80186
How to show that $K\leq S_4$ is a normal subgroup?
13742
Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
1220
Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
51359
Prove that every subgroup of an abelian group is a normal subgroup.
54911
The preimage of a normal subgroup under a group homomorphism is normal
146760
Prove that H is a subgroup of G
20386
Any subgroup of index $p$ in a $p$-group is normal.