Let $G$ be a finite group of order $m$. Does there exist a surjective group homomorphism from $GL(n, \mathbb{F}_{p}) \rightarrow G$ for some $ n \in \mathbb{N}$ and prime $p$?
2026-03-25 22:10:08.1774476608
Surjective group homomorphism from $GL(n, \mathbb{F}_{p})$
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According to this answer, the abelianization of $GL_n(\mathbb{F}_p)$ is (almost always) the multiplicative group of $\mathbb{F}_p$, and in particular it is always cyclic; so if $G$ is abelian but not cyclic, any morphism $GL(n, \mathbb{F}_{p}) \rightarrow G$ must factor through a cyclic group, and thus it cannot be surjective.