If $H$ is a subgroup of a group $G$ and $K$ is a normal subgroup of $G$ then show that $K$ is a normal subgroup of $HK$.
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.
90449
Showing that a Sylow subgroup of a group of order $p^{2}q$ is normal.
65037
$H$ is a maximal normal subgroup of $G$ if and only if $G/H$ is simple.
97833
Prove that if $k\mid m,$ then $Z_m$ has a subgroup of order $k.$
154398
Show that H is a subgroup of G
146760
Prove that H is a subgroup of G
65744
How can I Prove that H=G or K=G
61379
Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
95534
If $|G|=p^nq$, then $G$ contains a unique normal subgroup of index $q$
39165
Suppose $G$ is a group and $H,K \trianglelefteq G$. If $H \cap K =1$ the $hk=kh$ for all $k \in K, h \in H$?
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
44152
If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
102979
If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
5042
Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
42891
Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
44934
Show that every normal subgroup can be written as an intersection of $\ker \chi$'s for $\chi \in \operatorname{Irr}(G).$
102622
K is a subgroup of G. Show that K is not a normal subgroup