Show that $H$ is a normal subgroup of $G$ is it has index $2$.
75568
Subgroup of index $2$ is Normal
55350
H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
15630
Normal subgroup of prime order is characteristic
94999
A subgroup with an index of 2 is normal to the group.
7973
Groups of order $pq$ have a proper normal subgroup
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
44934
Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
36040
A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
1220
Have $\alpha:G \to H$. If a subgroup $U$ is normal in $H$ prove the pre-image of $U$ is normal in $G$.
22444
Normal subgroups of dihedral groups
25404
Let $G$ be a group of order $2m$ where $m$ is odd. Prove that $G$ contains a normal subgroup of order $m$
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite