$\DeclareMathOperator\SL{SL}$I would like to know if there is a full list of isomorphic class of finite subgroups of $\SL_5(\mathbb{C})$. I have already found the classification for $\SL_2$ and $\SL_3$. Thank you very much!
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2$\begingroup$ Incidentally, what is the classification for SL3? I only know the SL2 case. $\endgroup$– Theo Johnson-FreydCommented Jul 23, 2021 at 14:59
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5$\begingroup$ @TheoJohnson-Freyd: See H.F. Blichfeldt, The finite, discontinuous, primitive groups of collineations in three variables. Math. Ann. 63 (1907), no. 4, 552–572. $\endgroup$– Richard LyonsCommented Jul 23, 2021 at 16:02
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1$\begingroup$ I think the original result for SL3 is due to Camille Jordan digizeitschriften.de/dms/img/?PID=GDZPPN00215675X $\endgroup$– Abdelmalek AbdesselamCommented Jul 23, 2021 at 18:15
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2$\begingroup$ @AbdelmalekAbdesselam: Blichfeldt credits Jordan but points out that Jordan missed two perfect subgroups of $SL_3(C)$, namely $GL_3(2)$ and $3A_6$, of orders $168$ and $1080$. $\endgroup$– Richard LyonsCommented Jul 23, 2021 at 23:34
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1$\begingroup$ Since you did not restrict to irreducible groups, you also need the list of irreducible finite subgroups of ${\rm GL}(4, \mathbb{C})$ in addition to the groups given in Richard's answer. These groups were also know to Blichfeldt (at least the primitive ones). There were some omissions/errors, eg one found by Conway. $\endgroup$– Geoff RobinsonCommented Jul 24, 2021 at 15:43
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1 Answer
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The irreducible primitive finite subgroups of $SL_5(C)$ were classified up to isomorphism by Richard Brauer: "Über endliche lineare Gruppen von Primzahlgrad", Math. Ann. 169 (1967), 73–96.
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4$\begingroup$ By the way, believe that some of the groups in Brauer's list are actually imprimitive : for example, the $5$ dimensional character of $A_{5}$ is induced from any non-trivial linear character of $A_{4}.$ $\endgroup$ Commented Jul 23, 2021 at 16:02
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3$\begingroup$ @GeoffRobinson That is a slightly unusual example of an imprimitive group in that it acts faithfully on the blocks, so its kernel is trivial. $\endgroup$ Commented Jul 23, 2021 at 16:25
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2$\begingroup$ @GeoffRobinson: Thanks for the clarification. $\endgroup$ Commented Jul 23, 2021 at 23:39