Notation and principal results
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All groups are written additively unless a finite target group is explicitly allowed to be nonabelian. The natural numbers start at zero. Primes are indexed by
\[
p_0=2,\quad p_1=3,\quad p_2=5,\quad p_3=7,\quad\ldots.
\]
| Symbol | Meaning |
|---|---|
| \(C_p={\mathbb F}_p\) | The additive cyclic group of prime order \(p\). |
| \(D=\bigoplus_p C_p\) | Finite-support prime-coordinate vectors. |
| \(P=\prod_p C_p\) | All prime-coordinate vectors, with product topology. |
| \(q:P\to V\) | The quotient map to \(V=P/D\). |
| \(D_B\) | The subgroup supported on prime indices in \(B\subseteq{\mathbb N}\). |
| \(H_W=q^{-1}(W)\) | The intermediate group associated to a rational subspace \(W\le V\). |
| \(S_A\) | The prime set \(\{p_{2n+{\mathbf 1}_A(n)}:n\in{\mathbb N}\}\). |
| \(u_A\) | The coordinate indicator of \(S_A\) in \(P\). |
| \(G_A\) | The group \(q^{-1}({\mathbb Q} q(u_A))\). |
| \(s_A(r)\) | The specified rational lift of \(r q(u_A)\) to \(P\). |
| \(c_A(r,t)\) | The finite cocycle \(s_A(r)+s_A(t)-s_A(r+t)\). |
| \(e_p\) | The vector supported at \(p\) with value one. |
| \(p\mid_G g\) | There exists \(y\in G\) with \(py=g\). |
| \(A{=^{*}} B\) | The symmetric difference \(A\triangle B\) is finite. |
| \({\operatorname{rad}}(m)\) | The product of the distinct prime divisors of \(m\), with \({\operatorname{rad}}(1)=1\). |
| \(\widehat G\) | The profinite completion, not the Pontryagin dual. |
| \(A\le_T X\) | Membership in \(A\) is computable using oracle \(X\). |
| \(K_{\mathfrak G}(G)\) | Least admissible finite description length, or \(+\infty\). |
The main statements have the following form:
\[\begin{aligned}
{\operatorname{Tor}}(G_A)&=D,&G_A/D&\cong{\mathbb Q},\\
G_A/mG_A&\cong C_{{\operatorname{rad}}(m)},&\widehat{G_A}&\cong P,\\
G_A\cong G_B&\iff A{=^{*}} B,&&\\
G_A\text{ has an }X\text{-computable copy}&\iff A\le_T X.&&
\end{aligned}\]
The extension by \(D\) does not split. Distinct eventual selector classes do not embed in one another. All of these conclusions are proved within the book, with the exact hypotheses retained.