http://en.wikipedia.org/wiki/Symmetric_group#multiplication gives us (finally) background on the example in turings manuscript of squaring a permutation (the upright) to get two subcycles. Recall that he told crypto secrets (circa 1954) about wheel wiring, teaching such as how to then consider notions of distance of those factors and their relationship to primes and gcd. These are the conditions likely to identify his non exceptional groups – out of which he can then build a stochastic operator. Being entirely discrete, there seems no reason why in 1947 an r&d colossus cannot be computing such function based on high speed tape processing (still).
Gchq would do us a service by finding reaearch notes specifically characterizing post war colossus “research”, as it might then be seen in the light of computing history.
It feels like turing getting back at the uk establishment, for their oppresion of gays and the uk lackey status as one projecting the us pogrom. Upper class folks like turing were supposed be doing the same as his father did in the indian raj, helping oppress obamas father in kenya for example. Freethinkers and deniers are the worst subversive of all, in the uk, being given the henry the eight treatment in response – since they threaten the very foundation of the social order.
I can now see that his focus on automorphism groups was also a wheel wiring related disclosure (cloaked to hide it all from the censor). Without hitting censorship point, he argues in terms of alternating groups, families of which which relate to automorphism group of the free groups (trees!). This gets him to expander graphs and wheel wirings (or rather the means to search out the latter in reasonable time) – another use for post war colossus, perhaps. Its is these graphs, which the desired propeties, that go into early pseudo-random stream generators, for additive Ciphers woeking with quantizers of voice waveforms.