A normal subgroup $H$ of $G$, $[G:H] = n$
113925
Normal Subgroup Counterexample
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
80186
How to show that $K\leq S_4$ is a normal subgroup?
102622
K is a subgroup of G. Show that K is not a normal subgroup
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
23533
All subgroups normal $\implies$ abelian group
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
64269
Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
55493
Showing that the path-connected component of the identity matrix in a subgroup of $GL_n(\Bbb R)$ is a 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
50706
$|G|=p(p+1)$ for $p$ prime, then $G$ has a normal subgroup of order $p$ or $p+1$
116786
normal subgroups of Alternating group
131204
A non-trivial normal subgroup N of a finite group G or order $p^n$ is such that $N\cap Z(G)\neq \{ 1\}$
151727
If $|G/H|=4$ then $G$ is union of three proper subgroups
97928
Normal subgroup of a normal subgroup