normal subgroup of a group $G$
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
19458
Finite groups and Normal subgroups - $G$ is a finite group such that $|G|=n$. Let $p$ be the smallest prime such that $p|n$.
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
27703
an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
151699
Group of inner automorphisms of a group $G$
61373
A normal subgroup that is not a characteristic
67272
Deduce that if $G$ is a finite $p$-group, the number of subgroups of $G$ that are not normal is divisible by $p$
43651
Normal subgroups and factor groups
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 $G$
11957
G is the product of its Sylow subgroups
36040
A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
154878
Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
125244
Equivalent definitions of normal subgroups