Let $G=GL(4, 11)$ be the general linear group of $4\times 4$ matrices over the field $\mathbb{F}_{11}$. Let $H=SL(2, 7)$ be a special linear group of $2\times 2$ matrices over the field $\mathbb{F}_{7}$. I want to know whether there is any non-trivial group homomorphism $\phi$, where $$\phi:H\to G.$$ I know that order of $SL(2, 7)$ divides order of $G$ (or $|G|$). Therefore, such a non-trivial homomorphism may exist as $\phi(H)$ divides $|H|$ as well as $|G|$. Next, I know that any homomorphism from $H$ can be defined only from its generators. But I am not being able to find it. Please help.
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$\begingroup$ Why those specific groups? $\endgroup$– anonMar 19 at 21:47
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1 Answer
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Yes, there is an injective homomorphism with image generated by the matrices $$ \left(\begin{array}{cccc}10&6&0&0\\0&1&1&0\\0&10&0&0\\0&4&0&10\end{array}\right),\quad \left(\begin{array}{cccc}0&1&1&0\\0&0&4&1\\0&0&1&0\\1&0&6&0\end{array}\right).$$ There is even a nontrivial homomorphism ${\rm SL}(2,7) \to {\rm GL}(3,11)$ but that has image isomorphic to ${\rm PSL}(2,7)$.
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1$\begingroup$ Thank you very much for your brilliant answer. Can you please explain me how you have constructed this map? $\endgroup$– PAMGMar 19 at 17:02
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2$\begingroup$ It was a computer calculation, which you can do in GAP or Magma. I just computed the irreducible representations of ${\rm SL}(2,7)$ over ${\mathbb F}_{11}$, which is an easy calculation, and then wrote down two generating matrices from the image of a $4$-dimensional irreducible. $\endgroup$ Mar 19 at 19:06
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1$\begingroup$ The GAP command for this is
IrreducibleRepresentations(G,GF(11)). $\endgroup$– ahulpkeMar 19 at 20:24 -
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