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 → φk ∈ Aut(N) such that the group law in N o K is given by (n1k1)(n2k2) = n1 k1 n2 k2 = n1 φk1(n2) k1 k2
If 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 ∈ G
In 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 ∈ K
One 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 φk is the identity map on N, for all k ∈ K.
particularly when, later, it says
Recall that whenever we have an action of a group G on a space X, we get an action of G on functions on M by g · f(x) = f(g−1 · x)
In the case of a semi-direct product, φ gives us an action of K on the space N by k · n = φk(n) and so we get an action of K on the functions on N, and thus on Nˆ.
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.