- Joined
- Aug 5, 2015

- Messages
- 1

- Likes
- 0

Thread starter
#1

Hi. I'm trying to define the group of the Rubik's cube in arbitrary dimension. I shall explain what i figured out till now.

Let \( T_n \) be the group of rotational isometries of the n-dimensional cube. It acts transitively and faithfully on the
m-dimensional faces of the n-dimensional cube for any m between 0 and n.

Let
be the stabilizer of any of such faces (it does not depend, up to isomorphism, on the chosen face): that is, we consider the rotational isometries of the n-dimensional cube fixing a given m-dimensional face. So I define the 'illegal' n-dimensional Rubik's cube group to be:

That's because each of the m-dimensional pieces of the puzzle would have
as orientation group because that piece would be 'attached' by one of its m-dimensional faces (any piece is itself an n-dimensional cube) to the 'core' of the whole puzzle.

My problem is: i can't figure out how to define the generators on that group to build the real n-dimensional Rubik's cube group. It may be useful to note that by Krasner-Kalujnin embedding we have that \( T_n \) is embedded (in the way that we expect) in our illegal group.

Has any of you got any idea?

(sorry for the latex mess, i got problems on this forum using implemented latex tool)

Let \( T_n \) be the group of rotational isometries of the n-dimensional cube. It acts transitively and faithfully on the

Let

My problem is: i can't figure out how to define the generators on that group to build the real n-dimensional Rubik's cube group. It may be useful to note that by Krasner-Kalujnin embedding we have that \( T_n \) is embedded (in the way that we expect) in our illegal group.

Has any of you got any idea?

(sorry for the latex mess, i got problems on this forum using implemented latex tool)