Mathematics desk
< June 18	<< May \| June \| Jul >>	June 20 >

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

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)[reply]

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)[reply]

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)[reply]