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$SL_3(\mathbb{F_2})$ is the set of all invertible $3x3$ matrices over the field $F_2$ with determinant $1$. From what I understand $F_2$ is the set $\{0, 1\}$, ie.. $\mathbb{Z}$ mod $2$.

As far as I can see then, $SL_3(\mathbb{F_2})$ has to be $$I_3 = \begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}$$

If the $1$'s were in any other location we would have zero determinant. So is this $SL_3(\mathbb{F_2})$?

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    No. Look at the (other) permutation or upper/lower triangular matrices for instance. (Also note that the special and general linear groups are the same over $\Bbb F_2$, and $\times$ can be typeset with `\times`.) Why exactly do you think $1$'s anywhere else will cause zero determinant?2012-11-14

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Here is a complete list of total 168 elements of $SL(3,2) = SL_{3}(\Bbb{F}_2)$. I hope you may get some ideas on what is going on in $SL(3,2)$ from this.

enter image description here

(Or you can just play 'Where's Waldo?' game with Waldo replaced by the identity matrix $I_3$!)

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    But isn't don't the elements of $SL(3,2) = SL_{3}(\Bbb{F}_2)$ have to have $1$ as their determinant? Many of those matrices have $-1$ as a determinant?2012-11-14
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    By the way, how did you generate all those matrices...I'm guessing it wasn't by hand!2012-11-14
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    @sonicboom, In $\Bbb{F}_2$ we have $-1 = 1$. And I used *Mathematica 8.0* to find them all.2012-11-14
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    @sos440 +1 for the complete list2012-11-14
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    OK cheers. So over $\mathbb{F_2}$, $SL(3,2)$ has order $168$. But $GL(3,2)$ over $\mathbb{F_2}$ also has order $168$, so I take it over this particular field, the general linear subgroup and it's special linear group are equivalent?2012-11-14
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    @sonicboom, That is exactly what anon pointed out.2012-11-14
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    And @JayeshBadwaik, thanks for your precious upvote!2012-11-14
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    @sos440@: Ah yes I see he did mention it, thanks for the help.2012-11-14
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    I'll have to have a look into mathematica...I don't think matlab can do this kind of thing..ie outputting a list of elements of a group.2012-11-14