I'm trying to show that there's no subgroup of $ SL(2,5) $ isomorphic to $ A_5 $. I've already shown that $ A_5 $ is simple, and my strategy is to show that $ SL(2,5) $ contains no simple subgroups of order 60. My problem is that unfortunately I can't use Sylow theorems, which would make this a lot easier. Would anybody have any helpful tips/hints on how to start this WITHOUT Sylow theorems? Thanks!
2026-03-26 23:09:21.1774566561
Show that $ SL(2,5) $ has no subgroup isomorphic to $ A_5 $
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$SL(2,5)$ has only one element of order $2$ (which is $-I_2$), whereas $A_5$ has $15$ such elements (each being the product of two disjoints transpositions).