If $H$ and $K$ are finite supgroups of a group $G$, then $|HK| = \frac{|H||K|}{|H\cap K|}$
94469
How to prove that $|HK| = \dfrac{|H| \; |K|}{|H \cap K|}$?
27436
Does $G\cong G/H$ imply that $H$ is trivial?
15276
$G \times H \cong G \times K$ , then $ K \cong H$
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
97833
Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$
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.
63162
Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
2174
Proving that HK is a subgroup when K is normal
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
58033
$G\times H\cong G$ with $H$ non-trivial
86336
For $G$ group and $H$ subgroup of finite index, prove that $N \subset H$ normal subgroup of $G$ of finite index exists
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
105951
Does $G\times K\cong H\times K$ imply $G\cong H$?
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
110708
On proving that a finitely generated group has a finite number of subgroups with index $n$.