Show there is a normal subgroup $K$ of $G$ with $K\subset H$ and such that the order of $K$ divides $n!$
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
58251
To show every sylow p-subgroup is normal in G
19352
Normal subgroup of index m contains every element of order n where m and n are relatively prime
15630
Normal subgroup of prime order is characteristic
99999
A question on identifying normal subgroups of $S_4$
37379
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
129753
Prove that a subgroup which contains half of all elements is a normal subgroup.
37971
Prove that the center of a group is a normal subgroup
11731
Normal subgroups of $S_4$
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
22477
If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
50157
order of a sylow p-subgroup
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
13742
Proving that a normal, abelian subgroup of G is in the center of G if |G/N| and |Aut(N)| are relatively prime.
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.