ray5
Regular
- Joined
- Oct 10, 2020
- Messages
- 39
I have been investigating the PLL group a bit recently. By this I refer to the subgroup of the rubiks cube permutation group which permutes the top layer only, and keeps it oriented. This includes the identity, U, U2, U', the 21 PLLs but also every U conjugation of the PLLs.
How large is this group? I think the simplest way to count it is to use the fact that all permutations of the rubiks cube are even permutations, and then split PLLs into two cases: even * even and odd * odd type based on the way they affect corners and edges separately. The T perm is of odd * odd type and the G perms are of even * even type. There are 4!/2 = 12 even permutations on 4 points, and the same number of odd permutations. So in total the PLL group is of size 12^2 + 12^2 = 288.
It's well known that if you can do U moves, an A-perm and U-perm, you can build any PLL from just those. You can also get any PLL if you can just do U moves and Ga and Gc. Or Ra and Rb. Or Ja and Jb. Or T and Y.
What I found really interesting is that some pairs of PLLs are not enough to do full PLL with. If you can only do U moves and T-perm, you can do: Aa Ab Na Nb E H Z F T. If you can do U moves, an A-perm and E-perm then the only new perm you can get from combining these together is the H perm. Similarly Ua and H only lets you do Z. If you combine both N perms the only new perm you can make with them is the H perm.
In fact if you only have I moves and a single perm you can realize some very small groups in a new way. For example <U,Ja> and <U,Ra> are isomorphic to S4. <U,Na> is isomorphic to D8. <U,H> is isormophic to C4 x C2. So there is an interesting non-trivial subgroup lattice here.
The GAP scripts and output are stored in this repo https://github.com/river-000/gap-rubiks
How large is this group? I think the simplest way to count it is to use the fact that all permutations of the rubiks cube are even permutations, and then split PLLs into two cases: even * even and odd * odd type based on the way they affect corners and edges separately. The T perm is of odd * odd type and the G perms are of even * even type. There are 4!/2 = 12 even permutations on 4 points, and the same number of odd permutations. So in total the PLL group is of size 12^2 + 12^2 = 288.
It's well known that if you can do U moves, an A-perm and U-perm, you can build any PLL from just those. You can also get any PLL if you can just do U moves and Ga and Gc. Or Ra and Rb. Or Ja and Jb. Or T and Y.
What I found really interesting is that some pairs of PLLs are not enough to do full PLL with. If you can only do U moves and T-perm, you can do: Aa Ab Na Nb E H Z F T. If you can do U moves, an A-perm and E-perm then the only new perm you can get from combining these together is the H perm. Similarly Ua and H only lets you do Z. If you combine both N perms the only new perm you can make with them is the H perm.
In fact if you only have I moves and a single perm you can realize some very small groups in a new way. For example <U,Ja> and <U,Ra> are isomorphic to S4. <U,Na> is isomorphic to D8. <U,H> is isormophic to C4 x C2. So there is an interesting non-trivial subgroup lattice here.
The GAP scripts and output are stored in this repo https://github.com/river-000/gap-rubiks
