Is a subgroup $H$ of a group $G$ normal if $g^2 \in H$ for all $g \in G$?
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 $G$
12589
Show that every subgroup of the quaternion group Q is a normal subgroup of Q
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
139096
$|AxA|$ is a constant implies normal subgroup
55350
H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
11731
Normal subgroups of $S_4$
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
44165
the image of normal subgroups
47731
Intersection of all $p$-Sylow subgroups is normal
75568
Subgroup of index $2$ is Normal
7973
Groups of order $pq$ have a proper normal subgroup
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
36040
A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
129753
Prove that a subgroup which contains half of all elements is a normal subgroup.