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\section{Proof of the main theorems}\label{sec:proof}
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\Cref{table:summary} lists, for each of the $110$ groups of $\mathcal M_{10}$, where it is shown that it is not a
subgroup of the torsion of an elliptic curve over a number field of degree $10$.

\begin{table}[H]
\centering
\footnotesize
\begin{tabular}{@{}l@{\quad}l@{}}
\toprule
$G\in\mathcal M_{10}$ & treated in\\
\midrule
$\torz n$ for the $45$ levels $n$ of \Cref{table:global} & \Cref{prop:globaltable} (global method)\\
$\torz n$, $n=115,121,135$ & \Cref{prop:115-121,prop:135}\\
$\torz{155}$ & \Cref{prop:155} (descent)\\
$\torz n$, $n=95,111$ & \Cref{prop:95-111}\\
$\tg{11}{11}$ & \Cref{prop:11x11}\\
$\tg{3}{21}$, $\tg{3}{24}$, $\tg{4}{24}$ & \Cref{prop:3x21,prop:3x24-4x24}\\
$\tg{4}{16}$, $\tg{4}{20}$ & \Cref{prop:4x16,prop:4x20}\\
$\tg{3}{27}$, $\tg{3}{33}$, $\tg{4}{28}$ & \Cref{prop:3x27,prop:3x33,prop:4x28}\\
$\tg{3}{30}$, $\tg{3}{45}$ & \Cref{cor:3x30-3x45}\\
$\tg{3}{36}$, $\tg{6}{18}$ & \Cref{cor:3x36-6x18}\\
$\tg{3}{39}$, $\torz{78}$ & \Cref{prop:78-3x39}\\
$\tg{2}{32}$, $\tg{2}{34}$, $\tg{2}{36}$ & \Cref{prop:2x32,prop:2x34,prop:2x36}\\
$\tg{2}{38}$, $\tg{2}{40}$, $\tg{2}{50}$ & \Cref{prop:2x38-2x40-2x50}\\
$\tg{2}{42}$ & \Cref{prop:2x42}\\
$\torz n$, $n=64,68,72,76,80,84,100$ & \Cref{lem:isogeny} and \Cref{sec:2x2n}\\
$\torz{52}$ & \Cref{prop:52}\\
$\torz n$, $n=44,46,48,54,56$ & \Cref{prop:44,prop:46,prop:48,prop:54,prop:56}\\
$\torz n$, $n=58,60,62,66,70,74$ & \Cref{prop:58,prop:60,prop:62,prop:66,prop:70,prop:74}\\
$\torz n$, $n=49,51,55,75$ & \Cref{prop:49,prop:51,prop:55,prop:75}\\
$\torz n$, $n=77,85,91,93$ & \Cref{prop:77,prop:85,prop:91,prop:93}\\
$\torz n$, $n=57,63,65,81,87$ & \Cref{prop:57,prop:63,prop:65,prop:81,prop:87}\\
$\torz n$, $n=69,90,99,105,117$ & \Cref{prop:69,prop:90,prop:99,prop:105,prop:117}\\
\bottomrule
\end{tabular}
\medskip
\caption{Treatment of the $110$ groups of $\mathcal M_{10}$.}
\label{table:summary}
\end{table}

\begin{proof}[Proof of \Cref{thm:main}]
By \Cref{lem:base,prop:families,prop:explicit_cyclic,prop:2x28-2x30} and by
\Cref{prop:2x26,prop:3x15,prop:3x18,prop:6x12}, every group of $\mathcal G_{10}$ is the torsion group of an elliptic
curve over a number field of degree $10$. Conversely, let $E$ be an elliptic curve over a number field $K$ of
degree $10$. If $E(K)_\tors\notin\mathcal G_{10}$, then by \Cref{lem:minimal} it contains a group of $\mathcal M_{10}$,
which is impossible by the results listed in \Cref{table:summary}.
\end{proof}

\begin{proof}[Proof of \Cref{thm:sporadic}]
This is \Cref{prop:2x26,prop:3x15,prop:3x18,prop:6x12}.
\end{proof}