I’ve never understood, till now, the cryptographic significance of Turing (in his on permutations paper) of introducing the non ablelian quaternion group.
If u look at the four 1 dimensional representations of the quaternion groups, one has all four combinations of 2 bits.
When wheel breaking a tinny wheel, with its constraints on patters that influence which bit value in the sequence follows its prior, it’s crucial to cast such bit pairings and their biases in abstract terms, for which the quaternion groups 1-dimensional representations fit perfectly.
One has to recall how differential cryptography was conceived then (vs now).
Deltaing allowed convolutions of the “hermitian (quadratic) form” to amplify the bias, allowing reinforcement of biases when (delta’ed) streams are added
They thought of it all as a faltung (vs a generalized quadratic form) for which 2 binary digits are a special case.
Now if one was charged with finding machinery to assist in wheel breaking, being done manually but at high time cost, then noting that a system of suitably wired wheels (wired according to the quaternion group algebra) could represent an ideal set of likelihoods would give rise to it, when suitably biased as a detector of correct wheel patterns
That it all applied to enigma key/wheel breaking (as well as tunny) is still a secret!
Now one can see the algebra of bulges as an arithmetic of semi simple irreps, expressed in the quaternion group basis.
That the groups geometry happens to slightly with an affine simplex allowing limiting convergence of sequences of good guesses is also another secret