Let $H$ is a subgroup of $G$ and the index of $G\backslash H$ is 2. Prove that $H$ is a normal subgroup of $G$.
75568
Subgroup of index $2$ is Normal
123610
How to show that the commutator subgroup is a normal subgroup
44934
Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
20386
Any subgroup of index $p$ in a $p$-group is normal.
98542
Prove that $H$ is a subgroup of an abelian group $G$
54911
The preimage of a normal subgroup under a group homomorphism is normal
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 $G$.
99999
A question on identifying normal subgroups of $S_4$
118337
Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
71819
Prove that $H$ is normal.
144582
Index of a subgroup in $SL_2(\mathbb{Z})$
53409
Show group of order $4n + 2$ has a subgroup of index 2.
37971
Prove that the center of a group is a normal subgroup
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
58863
Proof that $A_n$ the only subgroup of $ S_n$ index $2$.
22444
Normal subgroups of dihedral groups