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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 $p$?

Sunny
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  • What is $m$? You introduced $m$ and then did not refer to it again. – Derek Holt Apr 06 '18 at 13:24
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    I guess the question is: What finite groups are homomorphic images of $GL(n, \mathbb{F}{p})$ ? This reduces to: What are the normal subgroups of $GL(n, \mathbb{F}{p})$? This seems easier to handle. – lhf Apr 06 '18 at 14:04
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    See for instance https://online.tugraz.at/tug_online/LV_TX.wbDisplaySemplanDoc?pStpSplDsNr=13888 – lhf Apr 06 '18 at 14:14

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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.

Arnaud D.
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  • if suppose $G$ is non-abelian group, then can you say something? – Sunny Apr 06 '18 at 13:10
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    I don't understand what you are asking. It depends on $G$. For most finite groups $G$ the answer is no. – Derek Holt Apr 06 '18 at 13:21
  • @SunnyRathore: It follows immediately that if $G$ is a quotient of a finite general linear group, it must have cyclic abelianization. – tomasz Apr 06 '18 at 14:07