In a group $G$, of order $p^k$ has a subgroup of order $p^{k-1}$ prove this subgroup is normal.
7108
Normal subgroup of prime index
23885
Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
111375
Normality of subgroups and Sylow groups under homomorphisms...
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
42389
If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
115204
Show that SO(n) is a normal subgroup of O(n)
113357
Groups of order $pqr$ and their normal subgroups
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
102622
K is a subgroup of G. Show that K is not a normal subgroup
35602
If a subgroup has smallest prime index, then it is normal
6134
Some Subgroup of Dihedral Group is Normal
7973
Groups of order $pq$ have a proper normal subgroup
11825
Let $H$ be a subgroup of $G$. Let $K = \{x \in G: xax^{-1} \in H \iff a \in H\}$. Prove that $K$ is a subgroup of $G$.
11731
Normal subgroups of $S_4$
25832
Let $K$ be a normal subgroup of order 2 in group $G$, show that $K$ lies in the centre of $G$