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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 ?

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  • $\begingroup$ Maybe determinant? $\endgroup$ – Willard Zhan Oct 31 '14 at 4:24
  • $\begingroup$ like the set of 2 by 2 matrices that maps to the determinant ? $\endgroup$ – alkabary Oct 31 '14 at 4:25
  • $\begingroup$ 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. $\endgroup$ – Willard Zhan Oct 31 '14 at 4:27
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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.

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How about a projection $S_3 \times \mathbb{Z}_2 \rightarrow \mathbb{Z}_2$ defined by $(\sigma, n) \mapsto n$.

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