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Normally, we think of Rubik's cube symmetry in terms of a particular position x.  The symmetry of x can be defined as Symm(x), where Symm(x) is the set all symmetries m in M such that x^m=x, and where M is the group of 48 symmetries of the cube.  By the closure property, Symm(x) is a subgroup of M, and there are 98 possible such subgroups.  I wish to expand this definition of symmetry to encompass an entire subgroup of the Cube group G.

The primary motivation for the expanded definition is to provide a standard mechanism to deal with the symmetry of subgroups of the Cube group G such as H=<U,R>.  I'm probably going to make things more complicated than they need to be.  However, I don't recall seeing a general discussion of group symmetries before, neither on Cube-Lovers nor on this forum.