if there are closed subgroups $H_1, H_2$ of $G$, with $H_1 \triangleleft H_2$, such that $H_2/H_1$ is isomorphic to $S$. Finally, we denote by $\exp G$ the exponent of a finite group $G$, namely, the smallest integer $e {\geqslant}1$ such that $g^e=1$ for every $g \in G$.

The $\ell$-adic and adelic failures
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We start by observing that it is enough to restrict our attention to the case $N=M$:

\[rem-NM\] Suppose that there is a constant $C{\geqslant}1$ satisfying $$\begin{aligned}
\frac{M^2}{\left[K_{M,M}:K_M\right]}\quad\text{divides}\quad C\end{aligned}$$ for all positive integers $M$. Then for any $N\mid M$, since $[K_{M,M}:K_{M,N}]$ divides $(M/N)^2$, we have that $$\begin{aligned}
\frac{N^2}{\left[K_{M,N}:K_M\right]}=\frac{N^2[K_{M,M}:K_{M,N}]}{[K_{M,M}:K_M]}\quad \text{divides} \quad \frac{M^2}{\left[K_{M,M}:K_M\right]},\end{aligned}$$ which in turn divides $C$.

Elementary field theory gives $$\begin{aligned}
\frac{N^2}{[K_{N,N}:K_{N}]}=&\prod_{\overset{\ell|N}{\ell\text{ prime}}}\frac{\ell^{2n_\ell}}{[K_{N,\ell^{n_\ell}}:K_N]}=\\
=&\prod_{\overset{\ell|N}{\ell\text{ prime}}}\frac{\ell^{2n_\ell}} {[K_{\ell^{n_\ell},\ell^{n_\ell}}:K_{\ell^{n_\ell}}]}\cdot\frac{[K_{\ell^{n_\ell},\ell^{n_\ell}}:K_{\ell^{n_\ell}}]} {[K_{N,\ell^{n_\ell}}:K_N]}=\\
=&\prod_{\overset{\ell|N}{\ell\text{ prime}}} \frac{\ell^{2n_\ell}} {[K_{\ell^{n_\ell},\ell^{n_\ell}}:K_{\ell^{n_\ell}}]} \cdot [K_{\ell^{n_\ell},\ell^{n_\ell}}\cap K_N:K_{\ell^{n_\ell}}]\end{aligned}$$ where $n_\ell={v_\ell}(N)$. To see why the first equality holds, recall that the degree $[K_{N, \ell^{n_\ell}} : K_N]$ is a power of $\ell$, so the fields $K_{N, \ell^{n_\ell}}$ are linearly disjoint over $K_N$, and clearly they generate all of $K_{N,N}$.

Let $\ell$ be a prime and $N$ a positive integer. Let $n:={v_\ell}(N)$. We call $$\begin{aligned}
A_\ell(N):=\frac{\ell^{2n}} {[K_{\ell^{n},\ell^{n}}:K_{\ell^{n}}]}\end{aligned}$$ the *$\ell$-adic failure* at $N$ and $$\begin{aligned}
B_\ell(N):=\frac{[K_{\ell^{n},\ell^{n}}:K_{\ell^{n}}]} {[K_{N,\ell^{n}}:K_N]}=[K_{\ell^{n},\ell^{n}}\cap K_N:K_{\ell^{n}}]\end{aligned}$$ the *adelic failure* at $N$ (related to $\ell$). 