# Onto Algebra homomorphism between group rings.

I have to determine onto $$F$$-Algebra map from group algebra $$FS_5$$ to $$M_4(F)$$ where $$F$$ is any finite field of characteristic $$2$$ and $$S_5$$ is symmetric group of degree $$5$$ generated by $$a=(1,2,3,4,5)~, b=(1,2)$$. I tried it as follows .

I define group homomorphism between $$S_5$$ and $$GL_4(F)$$as

$$a\rightarrow \left[ {\begin{array}{cc} 0& 0& 0& 1\\ 1& 0& 0& 1\\ 0& 1&0&1&\\ 0&0&1&1&\\ \end{array} } \right]$$ and

$$b\rightarrow \left[ {\begin{array}{cc} 0& 0& 0& 1\\ 0& 0& 1& 0\\ 0& 1&0&0&\\ 1&0&0&0&\\ \end{array} } \right]$$

Now this group homomorphism can be extended to Algebra homomorphism between $$FS_5$$ and $$M_4(F)$$. But I don’t know that this is onto map or not . Please suggest me that it is onto or not. One can suggest different map that make onto Algebra homomorphism. Thanks.