**@ Haven**

Are you familiar with the following result?

Given groups $\displaystyle G$ and $\displaystyle H,$ if $\displaystyle \phi:G\to H$ is a homomorphism with kernel $\displaystyle K$ and is onto, then $\displaystyle G/K\cong H.$

This is the theorem that __Swlabr__ is suggesting you should use to solve your problem.

**@ Swlabr, tonio**

The term “first isomorphism theorem” is not a standard one in the literature of group theory. Different books call the result above by different names. In *A Course in Group Theory* by John F. Humphreys, for example, the author refers to this result as the Homomorphism Theorem (reserving the term First Isomorphism Theorem for a different result). Most likely __Haven__’s book does not call the theorem above the First Isomorphism Theorem either. Instead of beating about the bush, it might have been more helpful to __Haven__ if you had just stated the theorem explicitly, as I have done.