Proposition 2.2

For every profinite group $J$ in universe $u$ and discrete topological group $N$ in universe $v$ equipped with a jointly continuous action of $J$ by group automorphisms, suppose every finite subset of $N$ generates a finite subgroup. Let $p,p_0$ be distinct natural primes. Assume that every quotient $J/U$ by an open normal subgroup $U$ of $J$ is solvable and that every $n\in N$ satisfies $n^{p^k}=1$ for some $k\in\mathbb N$. Let $J_0$ be a closed normal subgroup of $J$ whose index is exactly $p_0$. For $A\le J$, write $H^1(A,N)$ for the set of continuous maps $f:A\to N$ satisfying $f(xy)=f(x)(x\cdot f(y))$, modulo the equivalence relation $f\sim g$ if there is one $n\in N$ such that $g(x)=n^{-1}f(x)(x\cdot n)$ for every $x\in A$; its distinguished element is the class of the constant map $1$. Let $I_J(A,N)$ be the subset of $H^1(A,N)$ consisting of classes with a representative $f$ such that, for every $j\in J$, there exists $n_j\in N$ satisfying $j\cdot f(j^{-1}xj)=n_j^{-1}f(x)(x\cdot n_j)$ for every $x\in A$ for which $j^{-1}xj\in A$. This condition is imposed only on that intersection; it does not require $j$ to normalize $A$. The distinguished element of $I_J(A,N)$ is again the class of the constant map $1$. Then restriction defines an injective and surjective map $H^1(J,N)\to I_J(J_0,N)$, $[f]\mapsto[f|_{J_0}]$, and sends the class of the constant map $1$ to that same distinguished class in the target. Since $J_0$ is normal, the conjugate-membership qualification in the definition of $I_J(J_0,N)$ holds for every $j\in J,x\in J_0$. The target consists of classes having the specified invariance property, not all classes on $J_0$. The group $N$ may be infinite or trivial, and the finite-generation hypothesis includes the empty finite set. The index is a natural-number cardinal, which is defined as $0$ at infinite index; its equality to the prime $p_0\ge2$ therefore requires finite index and excludes $J_0=J$. Both primes exclude $0$ and $1$.