we just talked about this in university and i was curious about normal subgroups on 3x3.
i found a few obvious ones so far:
identity, whole group (haha)
edge flips, corner twists
edge permutations, corner permutations
(and of course you can link the ones that are disjunct and get a new one.)
are there any more? if no, is there a proof for this?
what about puzzles with only one type of pieces and fixed orientation (e.g. dino cube). do they have normal subgroups besides the identity and the whole group?