From this question, it is clear that the group of $3\times3$ matrices over $\mathbb{Z}_2$ acts transitively as well as faithfully over the set of $3$-tuples. But how can we conclude that the group is isomorphic to a subgroup of $S_7$? Why is not isomorphic to a subgroup of $S_8$? Do we use Cayley theorem here? Thanks beforehand.
2026-03-25 16:02:27.1774454547
Isomorphism between $GL_3(\mathbb{Z}_2)$ and a subgroup of $S_7$
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The set $\Bbb Z_2^3$ has $8$ elements, and $GL_3(\Bbb Z_2)$ permutes them. So in that sense, yes, $GL_3(\Bbb Z_2)$ acts like a subgroup of $S_8$.
But there are only seven elements of $\Bbb Z_2^3$ that $GL_3(\Bbb Z_2)$ actually does anything to (the action isn't transitive on the entire set of $3$-tuples, as you claim). So it acts as a subgroup of $S_7$ on those elements.