Since g was changed to K in other parts of the text, I keep wondering if I should be changing it above, too. It makes no sense to me as the original g(a) (the proportion of inputs that are a). Note how Turing (originally, before he changed funky-g to K) seemed to be stroking the g carefully – much as one strokes h-bar.
K – as entropy – IS increasing (being uncertainty induced by the injection of randomness during path walking and generator selection).
In this case, Q seems to be a delta-like function for tensoring almost – suggesting ways of including or excluding terms of a general sequence.
After all, it makes sense that we can choose an element of H1, and cycle it till one hits the identity group element. And, it makes sense that Q(of that cycled element) has value 1, in 1d measure.
Then big-g of x (i.e. K(x)) would be constant through each coset , being 1/4 in his quaternion example.
If this holds, then the next paragraph starts to make SOME sense. For then one is interest in the conditions that make translated cosets, where the translate comes from H (not H1), but the conjugated term from HI is still in H1. The condition on the generator exponents being zero seems to be a REQUIREMENT for this H1 and H coset regime ( when H1 is normal,).
When he then talks about the factor group (H/h1) being cyclic of order s, it now makes some sense
especially when his quaternion table example then goes on to note that when generated by k, one gets a 2-cycle in the evolution of the limiting function.
I can see how for a coset with 2 terms, in a non-abelian group, how the direction of the product gets one to 4 terms (with k and adjoint-k) being added to j and i. And of course, we know (once the burn-in is complete), one get to a constant K, for each coset. Each coset seems to be one line in the 7-point geometry.
But we can now see how k (and k’) generates H1:
We do seem to be cycling through the 4 cases of a pair (that happen to be called I, J, and their complements). One thinks of a 4-case truth table… each row of which identifies a line in the geometry, waiting for a generator to translate the point along the line in some direction of other (with quantum bit flip).