If $H$ is a normal subgroup of $G$, prove $g^{n} \in H$
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
33945
Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
8841
A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
71819
Prove that $H$ is normal.
2174
Proving that HK is a subgroup when K is normal
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$
102622
K is a subgroup of G. Show that K is not a normal subgroup
7575
Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
151621
$G$ a group and $H, K\mathrel{\unlhd}G$. Assuming that $H \cap K = \{1_G\}$ and $G = \langle H, K \rangle$, prove that $G \cong H \times K$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
51359
Prove that every subgroup of an abelian group is a normal subgroup.
93596
If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$.
91772
Proving that $A_n$ is the only proper nontrivial normal subgroup of $S_n$, $n\geq 5$
63780
to show characteristic subgroup is normal in G