We know now that the main purposes of measure theory is to add structure to a set so that one may start integrating (hopefully). Thus, when Turing imposes a L2 measure he does so so that he can start to convolve functions, working with group actions to define the (derived) functions such as automorphisms. Of course, automorphisms are just conjugations, where the “functional point” that is calculated is the conjugate of the “functional point” that it modifies. Of course, with permutation groups, “functional points” in measures spaces (that add structure to mere “sets” full of mere (non-functional) points) are just permutation cycles – which intuitively are bijections/functions anyways.

One thing that always bugged us – in the sense that it showed our ignorance – was Turing’s choice of the terms he used when introducing the inner product (g, k) . Can we draw parallels between his k and the K of the semi-direct notion described below? Various matters suggest we can.

Can we look at that through the following eyes?

G = N ⋊ K

This is an example of the following general construction

Definition(Semi-direct product).A semi-direct product G = N ⋊ K of two groups N and K is a group such that• As a set G = N × K and any element g ∈ G can uniquely be written as g = nk, with n ∈ N and k ∈ K.

• N is a normal subgroup of G, a condition that is written as N⊳ G

• There is a homomorphism φ : k ∈ K → φ

_{k }∈ Aut(N) such that the group law in N o K is given by (n1k1)(n2k2) = n1 k1 n2 k2 = n1 φ_{k1}(n2) k1 k2If you haven’t seen the definition of a normal subgroup before, it is that N is a normal subgroup of G iff n ∈ N ⇒ gng

^{−1 }∈ N, ∀n ∈ N, g ∈ GIn our semi-direct product case, this is only non-trivial for g ∈ K, so could be rewritten n ∈ N ⇒ knk

^{−1 }∈ N, ∀n ∈ N, k ∈ KOne can show that N⊳ G is equivalent to the existence of a group structure on the set of N cosets in G. Defining [gN] for g ∈ G to be the equivalence class where one identifies all elements of G of the form gn, for n ∈ N, when N⊳ G one gets a well-defined product on these equivalence classes, giving the quotient group G/N. One can always define the set G/N, but only when N⊳ G is this set a group.

Aut(N) is the so-called automorphism group of N, the group of homomorphisms from N to itself. Note that the product group N×K is the special case of N ⋊ K where φ

_{k }is the identity map on N, for all k ∈ K.Quantum Mechanics for Mathematicians- Semi-direct Products …

particularly when, later, it says

Recall that whenever we have an action of a group

Gon a spaceX, we get an action ofGon functions onMbyg·f(x) =f(g^{−1 }·x)In the case of a semi-direct product,

φgives us an action ofKon the spaceNbyk·n=φ(_{k}n) and so we get an action ofKon the functions onN, and thus onNˆ.Quantum Mechanics for Mathematicians- Semi-direct Products …

In his later arguments, is Turing making a semi-direct product argument? It would make quite some sense, if he is, particularly if the following statements about S are making statements about k controlling which permutation actions are generated.

What we have learned generally is that Turing, in the missing page, introduces a J, a group of 8 terms. It then gets cast as 8 drums, where the various mutual rotations of the drums represent the 8 symbols. This source language can written as a series of enigma drums in a cage, whose (constant) upright can be rotated. Being a language, it can “compiled” to (induced as) a set of operations on cosets, which have group structure too, enabling calculation. The uprights and initial stepping actions of the wheels are a further codification of J, more rigorous than analytic, and ensuring that one has a system of invariants – that must be preserved under unitary transforms. Similarly, one has a geometry that also codfies the invariants and the actions that result. This also induces expressions in the coset group, from which we get to the semi-direct claims.