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Conjecture: The order of the additive list always evenly divides the order of the generated group.

Time for some definitions.

First from the possible moves of the cube using Singmaster notation (U, D, F, B, L, R) pick any number of operators, this is the basis for the generated group.

The group generated by < U, F, D > is an example of the "generated group".

From these operators we can generate the "additive list" so the elements are

{ U, UF, UFD, UFDU, UFDUF, UFDUFD ... }

Now for some rules...