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This must be known to everyone who might remotely be considered an expert (but not to me...): What are the subgroups of index three of $SL(2, \mathbb{Z}/p \mathbb{Z})?$ EDIT Following up on Qiaochu's answer, the situation for $SL(2, 3)$ is described here. |
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A subgroup $H$ of index $3$ determines a homomorphism $\text{SL}_2(\mathbb{F}_p) \to S_3$ whose kernel $N = \bigcap_{g \in \text{SL}_2(\mathbb{F}_p)} gHg^{-1}$ is a normal subgroup of index either $3$ or $6$. The image of $N$ in $\text{PSL}_2(\mathbb{F}_p)$ is therefore also a normal subgroup of index $3$ or $6$. For $p \ge 5$ this group is simple and has size larger than $6$, so no such $N$ exists, hence no such $H$ exists. This leaves two cases $p = 2, 3$. When $p = 2$ we have $\text{SL}_2(\mathbb{F}_2) \cong S_3$ which has $3$ subgroups of index $3$ generated by each transposition. When $p = 3$ we have $|\text{SL}_2(\mathbb{F}_3)| = 24$, so its subgroups of index $3$ are precisely its Sylow $2$-subgroups. I think there are $3$ of them and that they are isomorphic to the quaternion group $Q_8$. (Edit: apparently there is only one.) |
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@Igor: If a group has subgroup of index 3, it has a homomorphism into $S_3$. |
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