For $H,K \triangleleft G$ and $H\cap K=\{1\}$ show $hk=kh \ \forall k \in K, h\in H$
23941
Normal Subgroups that Intersect Trivially
83273
Show that $o(gHg^{-1})=o(H)$
128834
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
128256
How to prove $H_1(\mathbb{C}/\Lambda, \mathbb{Z})$ to the lattice $\Lambda$ ,$\gamma \mapsto \int_\gamma dz$ is well defined?
98542
Prove that $H$ is a subgroup of an abelian group $G$
89659
$H_1 ,H_2 \unlhd \, G$ with $H_1 \cap H_2 = \{1_G\} $. Prove every two elements in $H_1, H_2$ commute
52105
Prove that $HK$ is a subgroup iff $HK=KH$.
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
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.
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
111129
Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.
58033
$G\times H\cong G$ with $H$ non-trivial
110217
$GL(n,K)$ is open in $M(n,K)$
80996
A canonical construction proving $\left|HK\right| = \left|H\right|\left|K\right| / \left|H \cap K\right|$?
67772
How to find $k$ such that $(\mathbb{Z} \times \mathbb{Z})/ \langle (m,n)\rangle \cong \mathbb{Z} \times \mathbb{Z}_k$
113013
Let $M$ and $N$ be two normal subgroups of $G$. Show that $M \cap N$ is also normal in $G$. Then show that $G/(M \cap N) \cong (G/M) \times (G/N)$