We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl}
\forall i & S_i \circ S_i & = \operatorname{Id}_E \\
\forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0
\end{array} \right.$$ We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Let $n$ be an odd integer. Prove that $p(n) = 1$.
We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that
$$\left\{ \begin{array}{lrl}
\forall i & S_i \circ S_i & = \operatorname{Id}_E \\
\forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0
\end{array} \right.$$
We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$.\\
Let $n$ be an odd integer. Prove that $p(n) = 1$.