Onto Algebra homomorphism between group rings.
$begingroup$
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
$
arightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
1& 0& 0& 1\
0& 1&0&1&\
0&0&1&1&\
end{array} } right]
$ and
$brightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
0& 0& 1& 0\
0& 1&0&0&\
1&0&0&0&\
end{array} } right]$
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.
ring-theory finite-groups modules group-rings free-modules
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
$endgroup$
add a comment |
$begingroup$
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
$
arightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
1& 0& 0& 1\
0& 1&0&1&\
0&0&1&1&\
end{array} } right]
$ and
$brightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
0& 0& 1& 0\
0& 1&0&0&\
1&0&0&0&\
end{array} } right]$
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.
ring-theory finite-groups modules group-rings free-modules
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
$endgroup$
add a comment |
$begingroup$
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
$
arightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
1& 0& 0& 1\
0& 1&0&1&\
0&0&1&1&\
end{array} } right]
$ and
$brightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
0& 0& 1& 0\
0& 1&0&0&\
1&0&0&0&\
end{array} } right]$
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.
ring-theory finite-groups modules group-rings free-modules
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
$endgroup$
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
$
arightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
1& 0& 0& 1\
0& 1&0&1&\
0&0&1&1&\
end{array} } right]
$ and
$brightarrow left[ {begin{array}{cc}
0& 0& 0& 1\
0& 0& 1& 0\
0& 1&0&0&\
1&0&0&0&\
end{array} } right]$
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.
ring-theory finite-groups modules group-rings free-modules
ring-theory finite-groups modules group-rings free-modules
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
edited Dec 30 '18 at 4:30
neelkanth
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
asked Dec 29 '18 at 15:23
neelkanthneelkanth
2,30521129
2,30521129
add a comment |
add a comment |
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