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An automorphism (specifically, a group automorphism) is simply an isomorphism that is from a group to itself. Most authors define inner automorphisms and outer automorphisms roughly as follows.

An inner automorphism is an automorpism of the form p(g)=G^h=h'gh for all g in G and for a specific, fixed h in G. An outer automorphism is an automorphism that is not inner.

But occasionally the definition takes a slightly different form. The alternate definition says that automorphisms are of the form p(g)=G^h=h'gh for all g in G. If h is a fixed element of G, then p is an inner automorphism. Otherwise, h is not in G but rather is a fixed element of a larger group of which G is a subgroup, and p is an outer automormphism. The latter definition clearly motivates the names "inner" and "outer".

I have been trying to enumerate all LL algorithms, (following Bernard Helmstetter's work) and cannot seem to reduce the 62208 permutations to the 1211 he has come up with. I am unaware of whether or not he used antisymmetry for his reductions. Using GAP and U rotations and reflections I have come up with a total of 8020. Initially considering antisymmetry as well (removing order 2 permutations and the identity from consideration. Note: I did not consider antisymmetric positions with dihedral symmetry or reflections) still does not bring it close to 1211. I was using Martin Schoenert's method in GAP and a very slow algorithm I wrote to determine equivalence classes from conjugation with a group (automorphism) separately with the same results. If anyone would like to see my definitions I will post them.

I have been dabbling with the numbers and here is what I have come up with: Let's assume that Moore's law will hold for another X years. This is not unreasonable following advancements in parallel processor designs, quantum and biological computing. So, in X years according to this model, computers could have the potential computing power of N=2^(X/1.5) times the power of todays computers. If a respectable computer from today could optimally solve (considering the optimal solvers of today) a cube in 30 minutes (1800 seconds)...then a computer from this possible future could solve the cube in about 1800/N seconds. Now, assuming we were to hack away at this until these God computers came about, we would only net the equivalent of twice the computing time on these new computers, or about 36 months. This means that to solve the cube completely by time T in years, we would need to solve around 4.8 billion cubes/second. If we also consider farming this out to other computers of equivalent capabilities and enumerating the reduced cube space of 450541810590509978, then we can solve for a potential time frame on exhaustively searching the cube space.