# second isomorphism theorem

Let $(G,*)$ be a group. Let $H$ be a subgroup of $G$ and let $K$ be a normal subgroup of $G$. Then

• $HK:=\{h*k\mid h\in H,\ k\in K\}$ is a subgroup of $G$,

• $K$ is a normal subgroup of $HK$,

• $H\cap K$ is a normal subgroup of $H$,

• There is a natural group isomorphism $H/(H\cap K)=HK/K$.

The same statement also holds in the category of modules over a fixed ring (where normality is neither needed nor relevant), and indeed can be formulated so as to hold in any abelian category.

Title second isomorphism theorem SecondIsomorphismTheorem 2013-03-22 12:08:46 2013-03-22 12:08:46 djao (24) djao (24) 9 djao (24) Theorem msc 13C99 msc 20A05