We can look at Turing’s schoolboyish appreciation of higher math, as it leads to notions of programming.
One sees why Turing would be good at codes, as he was first taught to reason geometrically – rather than having to study notions of generators, spanning sets, bases etc. from linear algebra. We can see now that to a Turing, the notion of an automorphism group is akin to a software program – specially when you take the semi-direct product of the group and the automorphism group (think input, and program). To a Turing, the automorphism group is just a set of card index organized to translate English to Chinese using an English to German and then a German to Chinese dictionary.
Similarly, the notion of foundation group or “Structural subgroup” is particularly evident in dihidral groups, which so clearly factor – as one breaks up factor chains of subgroups. The notion of the center then becomes also a programming-style concept – that which is entirely self-describing and self-sufficient. One can see also the notion of basis, when one then considers quotient groups.
Clearly, we can imagine re-wiring 8 of the 26 positions on an enigma “drum” according to the picture above; and we can imagine stringing several such rotors in a rotor array. From Turings own analysis, one looks at the array as a composite operator, reducing the variance-distance between the relative proportions fond in the input alphabet to the uniform proportion.
Something about the next 2 pictures totally makes the point that ifs fine to relabel things (using your card indexes). And indeed, a suitable re-labeling really shows up certain underling cycles:
Getting to the idea that an operation and a code for that operation can be seen as one and the same thing (so one has operations on numbers – which are operations) is obvious next (without resorting to more advanced von neumann linear operators, etc)
It seems pretty clear that Turing was “well taught” quaternions – not as some abstruse branch of math but as a reasoning system for calculating.