Let $F$ be any field.
Verify the group axioms for the special linear group $SL_n(F)$ whose elements are $n$ x $n$ invertible matricies with entries in $F$ and the product is matrix multiplication.
\begin{equation} SL_n(F) = [A \in GL_n(F) : \text{det} A =1]. \end{equation}
I have managed to verify the 3 axioms for $GL_n(F)$ but I am unsure how to extend this to $SL_n(F)$ . Any help would be appreciated.
We can "extend" this to $SL_n(F)$ as follows. After having seen that $GL_n(F)$ is a group we show that $$ \det\colon GL_n(F)\rightarrow F^{\times} $$ is a group homomorphism with kernel $SL_n(F)$. As a kernel, $SL_n(F)$ then is a normal subgroup of $GL_n(F)$, hence in particular a group. Of course, this amounts to showing the group axioms, because the fact that $\det$ is a group homomorphism means that $\det(AB)=\det(A)\det(B)$ (see the hints).