Testwiki:Reference desk/Archives/Mathematics/2017 June 19

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June 19

Solvability of the Rubik's cube group

I don't mean solvability of the puzzle, but rather, whether the group is a solvable group. To me, it seems like it is not solvable for similar reasons as the Galois groups involved in the Abel-Ruffini theorem. I've tried looking this up but couldn't find an answer.--Jasper Deng (talk) 05:45, 19 June 2017 (UTC)

All subgroups of a solvable group are solvable. But $A_8$ (even permutatiions of corner pieces) and $A_{12}$ (even permutations of edge pieces) are subgroups of the Rubik's Cube group, and $A_8$ and $A_{12}$ are not solvable (since they are simple and not Abelian). So we can conclude that the Rubik's Cube group is not solvable. Gandalf61 (talk) 09:17, 19 June 2017 (UTC)

Technically, the piece permutations form a homomorphic image of the Rubik group. But the homomorphism splits so there are isomorphic copies of these groups as subgroups. It should be mentioned that the Jordan–Hölder theorem comes into play here as well: the fact that if one non-solvable group appears as a quotient then there is no decomposition series with solvable factors. --RDBury (talk) 15:40, 20 June 2017 (UTC)