Suppose $H \le G$ and $g^2\in H$ for all $g\in G$. Show $H$ is a normal subgroup of $G$
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
2174
Proving that HK is a subgroup when K is normal
80186
How to show that $K\leq S_4$ is a normal subgroup?
43651
Normal subgroups and factor groups
103216
The set $\{g^2 | g \in G\}$ in a group $G$
45580
Intersection of conjugate subgroups is normal
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
155054
$G$ be a group and $H$ be a normal subgroup of index $p$ ( a prime ) ; suppose $K$ is a subgroup of $G$ not contained in $H$ , then $G=HK$?
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
78580
Is $SL_n(\mathbb{R})$ a normal subgroup in $GL_n(\mathbb{R})$?
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
123677
Normal subgroups of infinite symmetric group
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
116786
normal subgroups of Alternating group
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$