Let $H \triangleleft G$ such that $[G:H]=n$. Show that $g^n \in H$ for all $g \in G$.
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
104855
Prove that $N \subset Z(G)$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
141043
$H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
22477
If H is a subgroup of a group G and K is a normal subgroup of G the show that K is a normal subgroup of HK.
15276
$G \times H \cong G \times K$ , then $ K \cong H$
40094
Does $G\oplus G \cong H\oplus H$ imply $G\cong H$ in general?
34724
For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?
154398
Show that H is a subgroup of G
103216
The set $\{g^2 | g \in G\}$ in a group $G$
49166
$R^n \cong R^m$ iff $n=m$
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
37379
Let $G$ a group with normal subgroups $M,N$ such that $M\cap N=\{e\}$. Show that if $G$ is generated by $M\cup N$ then $G\cong M \times N$.
141683
$ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
43645
Show that $G/H\cong S_3$