I learned that the special linear group is a normal subgroup of the general linear group. That is, $SL_n(\textbf F) \lhd GL_n(\textbf F)$. It is quite obvious since $\det(ABA^{-1})=\det(B)=1$ where $A \in GL_n(\textbf F)$ and $B\in SL_n(\textbf F)$. I wonder about what does the quotient group $GL_n(\textbf F)/SL_n(\textbf F)$ equal to? or just about the order of this quotient group. I need your help.
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4Hint: $SL_n(F)$ is the kernel of a certain homomorphism from $GL_n(F)$ to $F^{\times}$. Which map and is it surjective? – Tobias Kildetoft Mar 24 '14 at 12:55