# Non abelian groups mapping to abelian groups

I want to find a non abelian group and a mapping $\phi$ that maps to a an abelian group , i want $\phi$ to be an epimorphism meaning that it is both a homomorphim and onto. I am thinking of a matrix group because they are famous for being non-abelian,but i can't find any mapping that is an epimorphim to an abelian group . any suggestions ?

• Maybe determinant? – Willard Zhan Oct 31 '14 at 4:24
• like the set of 2 by 2 matrices that maps to the determinant ? – alkabary Oct 31 '14 at 4:25
• The set of $2\times 2$ matrices is not a group. You can think of the set of $2\times 2$ invertible matrices. But for me, the answer of Tamaroff is essential. – Willard Zhan Oct 31 '14 at 4:27

Consider the mapping $\eta:G\to G/[G,G]$, where $G/[G,G]$ is the abelianization of $G$. This is the most basic example, and every other mapping of the sorts factors through this mapping.
How about a projection $S_3 \times \mathbb{Z}_2 \rightarrow \mathbb{Z}_2$ defined by $(\sigma, n) \mapsto n$.