This action induces the homomorphism $\phi:GL_2(\mathbb{Z}_2)\to S_4$, which is injective. Would it be correct to explicitly list the elements of $im(\phi)$ as the matrices in $GL_2(\mathbb{Z}_2)$?
2026-04-01 03:42:41.1775014961
$GL_2(\mathbb{Z}_2)$ acting on $\mathbb{Z}_2 \times \mathbb{Z}_2$
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Image $\mathrm{Im}\phi=\phi(A)=\{\phi(a)\}_{a\in A}=\{b\in B\ |\ \ \exists a\in A: \phi(a)=b \}$ of the homomorphism $\phi: A\to B$ is a subgroup of $B$. So if you map a group of matrices to a permutation group, then the elements of $\mathrm{Im}\phi$ are permutations. So no, it wouldn't be correct to list the elements of $\mathrm{Im}\phi$ as matrices, but it would be correct to list them as permutations.