Let $G$ be a group of odd order, and let $H$ be a subgroup of $G$ of index $5$. Prove that $H$ is a normal subgroup of $G$.
49666
Generalization of index 2 subgroups are normal
123677
Normal subgroups of infinite symmetric group
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
12589
Show that every subgroup of the quaternion group Q is a normal subgroup of Q
134353
A group of order $108$ has a proper normal subgroup of order $\geq 6$.
15526
Question on groups of order $pq$
108250
A group of order $595$ has a normal Sylow 17-subgroup.
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
103423
Normal subgroup where G has order prime
89508
Existence of normal subgroups for a group of order $36$
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
65972
Show that a group of order $108$ has a normal subgroup of order $9$ or $27$.
43651
Normal subgroups and factor groups
3439
Other proofs that subgroups of $A_5$ have order at most 12
1220
Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.