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Is it true that the group $G = \mathrm{GL}_2(\Bbb F_3)$ is a quotient of a dihedral group $D_{2n}$ for some $n$ ?

More generally, can $G$ be the quotient of a group $H$ which fits in an exact sequence $0 \to A \to H \to A' \to 0$ where $A,A'$ are finite abelian groups ? (We could also ask the same question for $G = \mathrm{GL}_2(\Bbb F_p)$ in general...)

I know that $G$ is solvable, so it might be related to extensions of abelian groups. I know that there is an exact sequence $0 \to \Bbb F_3^* \to G \to \mathrm{PGL}_2(\Bbb F_3) \cong S_4 \to 1$ (NB : $\mathrm{PGL}_2(\Bbb F_3)$ acts on the set $X$ of lines in $\Bbb F_3^2$, which has cardinality $4$). But I'm not sure about my precise question. Thank you!

Alphonse
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  • Is $G$ generated by two elements $a,b$ with $a^n = b^2 = (ab)^2 =1$ ? – reuns Sep 28 '17 at 16:50
  • @reuns : thank you for your comment. I'm not sure how I should solve your question, however. Maybe working with $G/Z \cong S_4$ ? Could you please provide some more help? – Alphonse Sep 28 '17 at 17:00
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    For generators of $G$ see this question. – Dietrich Burde Sep 28 '17 at 18:59
  • No, because it does not have a cyclic subgroup of index at most $2$, while every quotient of a dihedral group has this property. – verret Sep 29 '17 at 01:27
  • @verret : thank you very much! Do you know if $G$ can be the quotient of a group $H$ which fits in an exact sequence $0 \to A \to H \to A' \to 0$ where $A,A'$ are finite abelian groups ? I know that $H$ is solvable in that case, but since $G$ also is, it doesn't tell me it's impossible for $G$ to be a quotient of $H$ – Alphonse Sep 29 '17 at 05:56
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    No, in fact $G$ is not even metabelian. The only abelian normal subgroups of $G$ are the identity, and the center, and the quotient by the center is isomorphic to $S_4$. – verret Sep 29 '17 at 07:34
  • @verret : very nice. Thank you! You could maybe provide an answer (with some references for normal subgroups of $GL_2(\Bbb F_p)$, if you have – I know that $PSL_n(\Bbb F_p)$ is simple for $p>3,n \geq 2$, but I don't know to find all the normal subgroups of $GL_2(\Bbb F_p)$). – Alphonse Sep 29 '17 at 15:20

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The derived subgroup of a dihedral group is cyclic, and this property is inherited by quotients. On the other hand, the derived subgroup of $GL(2,3)$ is $SL(2,3)$, which is not even abelian. (In particular, $GL(2,3)$ is not even metabelian.)

verret
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