Show that the group of 3x3 invertible matrices over $\mathbb{Z}_2$ acts on the set of 3-tuples over $\mathbb{Z}_2$ by matrix multiplication. There are 8 such 3-tuples. Show that one of these are fixed by all matrices and that the group acts transitively on the others. Use these facts to prove that $GL_3(\mathbb{F}_2)$ is isomorphic to a subgroup of $S_7$.

If we treat the rows and the columns as tuples, it is clear to see that they are elements of $(\mathbb{Z}_2)^3$. I am not sure what "fixed" means in this context. I am also confused to how I could use these results to prove that there exists as isomorphism between $GL_3(\mathbb{F}_2)$ and a subgroup of $S_7$. Any hints as to how to continue would be much appreciated.

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    $\begingroup$ "Fixed" means that $g.x = x$ (for all $g$). or the second part, you get a map from your group to $S_7$ so you just need to show this is injective. $\endgroup$ – Tobias Kildetoft Nov 4 '16 at 8:02
  • $\begingroup$ @TobiasKildetoft By the wording of the question, it seems like it's asking for which tuple is fixed? Could you explain what "one of these" is referring to? $\endgroup$ – ultrainstinct Nov 4 '16 at 8:09
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    $\begingroup$ "one of these" refers to one of the tuples. Which one should be very easy to see once you write up the action. $\endgroup$ – Tobias Kildetoft Nov 4 '16 at 8:11
  • $\begingroup$ @TobiasKildetoft Yes, it is clearly the 0-tuple, thank you for that. What does transitively mean in this context? It doesn't make sense interpreting it in the traditional sense, where if x and y are related and y and z are related then x and z are related. $\endgroup$ – ultrainstinct Nov 4 '16 at 8:16
  • $\begingroup$ @Huffman_Coding that is not what transitive action means. See en.wikipedia.org/wiki/Group_action#Types_of_actions $\endgroup$ – user259242 Nov 4 '16 at 8:17

Transitivity in the context of group actions on a space $X$ means that any point of $X$ can be taken to any other. If suffices to prove that the vector $e_1=\begin{pmatrix}1\\0\\0\end{pmatrix}$ can be taken to any other nonzero vector $v$. This can be done by including $v$ as the first column of a nonsingular matrix.

  • $\begingroup$ You mean first row right? Why is it enough to prove it only for that vector? Does it follow that all other vectors can also be taken to any other vectors? How does this show there is an injection to $S_7$? Like what I mean is, what do these results have anything to do with 3x3 invertible matrices being isomorphic to a subgroup of $S_7$? $\endgroup$ – ultrainstinct Nov 4 '16 at 8:38
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    $\begingroup$ Since we are in a group, we can always invert transformations. So if we want to take $v$ to $w$, it suffices to take $v$ to the a standard vector $e_1$ first, and then $e_1$ to $w$. $\endgroup$ – Mikhail Katz Nov 4 '16 at 8:51
  • $\begingroup$ About my other questions, how is this relevant to showing that 3x3 invertible matrices are isomorphic to a subgroup of $S_7$? Can we just label each of the seven tuples 1 through 7, and then based on what a given matrix does to these tuples, transform that to a permutation? Is that sufficient? $\endgroup$ – ultrainstinct Nov 4 '16 at 9:13
  • $\begingroup$ Matrix multiplication is associative. Therefore what is defined above is a group action. $\endgroup$ – Mikhail Katz Nov 4 '16 at 9:16
  • $\begingroup$ Ohhhhhhh, ok I think I get it now. Thank you so much! $\endgroup$ – ultrainstinct Nov 4 '16 at 9:18

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