If $H$ and $K$ are subgroups of $G$, and $K$ is normal, then $HK$ is a subgroup of $G$
2174
Proving that HK is a subgroup when K is normal
116786
normal subgroups of Alternating group
1251
If $[G:K]$ is finite, then $[H:H \cap K] = [G:K]$ iff $G = HK$ (Hungerford Proposition 4.8, Proof)
34902
If $H$ is a cyclic subgroup of $G$ and $H$ is normal in $G$, then every subgoup of $H$ is normal in $G$.
70373
Intersection of normal subgroups proof
154398
Show that H is a subgroup of G
121587
Let $H$ be a normal subgroup of index $n$ in a group $G$. Show that for all $g \in G, g^n \in H$
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
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 $G$
142739
Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$
57227
Maximal normal $\pi$-subgroups and torsion subgroups
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
130455
If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
145511
Suppose $H$ is the only subgroup of order $o(H)$ in the finite group $G$. Prove that $H$ is a normal subgroup of $G$.
125244
Equivalent definitions of normal subgroups
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.