At page 23 we see turings argument (avoid n = 4, in the even permutations group) – since it has a subgroup of its own.
We also see, more generally, how the groups of fixed point compare with the orbits resulting form subgroups, acting much like the hamming parity matrix does in “constraining” solutions.
Looking at all this through the eyes of one designing a theory for wheel wirings its obvious that one can count the crossover of wires (going from one sides pads to the other sides pins). And its obvious that the organization of the wiring can be arrange such that one had the minimum number of crossings. The even parity idea (before binary took over) relates to these crossings – codified in the theory of the alternating group. Powers of (-1) gives positive/negative values to this notion of parity, with -1 to the kth power being because there are k crossings in this “1 dimensional” wheel. In a 2 dimensional space (ie 2 tunny chi wheels, each providing a value) we have “2-dimensional crossings” and thus (-1) to the power of the inner product of x1 and x2 (where one imagines the smaller dimension’s vector projecting onto the larger).
So turing gets from three cycle enumerations to non-exceptional groups, meaning every possible function can be a generator, where each is an element of An (because it has no factors when t>5).
The paper does a proper job of teaching why we want to know about stabalizers and kernels, normalizers (and a maximum normal subgroup), and centralizers and the center (from which one gets to conjugacy classes). It also clears up why an argument of that era wants to use cosets and associated group quotients – since it supports counting arguments and, more generally, it sorts orbits and classes alike into electron-cloud like orbits.
It was particular illuminating to see how a representation map roe can not only be a mapping from group element “function” to permutation (in cycles) that actually calculates (that “function“ in the sense of a subroutine); but can also be a map from the same named group element function to an automorphism, alternatively.
Now we have the background turing expected us to have and to apply to his minimalist argument that presumed that one would recognize a foundational point from the mere use of a technical term, or a supporting argument from foundations when correctly using technical terms in concert (with their associated allusion to the axiom at issue).
A good example is that of using an e to the pi….. function, as a basis. All this is really doing is using conjugacy classes ( each an (non commutative) orbit around the commutative enter) as the basis.
Interesting to get a glimpse into the very mindset of a pre-constructionist thinker, involved in early machine ciphering.
I can see now that just as there be might be switching basis in a series of bases that provides a means to direct how a cipher algorithm evolves, so alternatively one might be using the sign of the position one finds oneself on a walk through the alternating group to help decide, at the walk of a multi-particle system now, whether one chooses x or y path, for each particle.
That this all hinges on the parity (+1, -1) seems all the more likely when one can generalize to r generators – providing n and r have gcd of 1 (ie are relatively prime). Now we have a class!