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.


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