$|G|<\infty$. $p$ the smallest prime dividing $|G|$. $|G:H|=p$. then H is normal.
7108
Normal subgroup of prime index
71819
Prove that $H$ 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 $G$.
86594
If $x^2 \equiv y^2 \pmod {p^r}$, where $p$ is an odd prime not dividing $x$ or $y$ then $x \equiv \pm y \pmod {p^r}$
58251
To show every sylow p-subgroup is normal 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
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
20386
Any subgroup of index $p$ in a $p$-group is normal.
99630
For every $g\in G$ there exists an $h\in G$ such that $g = h^3$
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
94507
Given $G$ group and $H \le G$ such that $|G:H|=2$, how does $x^2 \in H$ for every $x \in G$?
7973
Groups of order $pq$ have a proper normal subgroup
33714
$p$ prime, $p\mid a^k \Rightarrow p^k\mid a^k$
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
112045
A property of minimal prime ideal
139096
$|AxA|$ is a constant implies normal subgroup
119512
If H ≤ Z(G) ≤ G, where G is a finite group,Z(G) is its center, and (G:H) = p for some prime p, then G is abelian.