Let $q\geq 5$ and let PGL(2,q) be the projective general linear group.
Question
Do there exists a $q$ such that PGL(2,q) is solvable?
2026-03-27 07:11:39.1774595499
Projective linear group - solvable
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$PSL(2,q)$ for all $q>3$ is simple and non-abelian, hence can not be solvable (since $G'=[G,G]$ is a normal subgroup). Since $PSL(2,q)\leq PGL(2,q)$, $PGL(2,q)$ can not be solvable either.