Suppose that H is a subgroup of K. Then $ (G:H)=(G:K)(K:H)$
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
85129
If $H_1, H_2\leq G$ are such that $H_1\cong H_2$ then $G/H_1\cong G/H_2$?
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
11957
G is the product of its Sylow subgroups
15276
$G \times H \cong G \times K$ , then $ K \cong H$
102979
If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
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.
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
44942
A group $G$ with a subgroup $H$ of index $n$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
105951
Does $G\times K\cong H\times K$ imply $G\cong H$?
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
106927
How do I prove $[G:H\cap K]\leq [G:H][G:K]$?
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$