We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. We assume here that $n$ is even and we set $n = 2m$. We consider:
an H-system $(S_1, \ldots, S_p, T, U)$ of $E$,
the subspace $E_0 = F_T = \operatorname{Ker}(T - \mathrm{Id})$,
for $j \in \llbracket 1, p \rrbracket$, the endomorphism $R_j = \mathrm{i} U \circ S_j$ of $E$.
a) Show that, for all $j \in \llbracket 1, p \rrbracket$, the subspace $E_0$ is stable under $R_j$. b) For $j \in \llbracket 1, p \rrbracket$, let $s_j$ be the endomorphism of $E_0$ induced by $R_j$. Show that $(s_1, \ldots, s_p)$ is an H-system of $E_0$. c) Deduce that $p(2m) \leqslant p(m) + 2$.
We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$.\\
We assume here that $n$ is even and we set $n = 2m$. We consider:
\begin{itemize}
\item an H-system $(S_1, \ldots, S_p, T, U)$ of $E$,
\item the subspace $E_0 = F_T = \operatorname{Ker}(T - \mathrm{Id})$,
\item for $j \in \llbracket 1, p \rrbracket$, the endomorphism $R_j = \mathrm{i} U \circ S_j$ of $E$.
\end{itemize}
a) Show that, for all $j \in \llbracket 1, p \rrbracket$, the subspace $E_0$ is stable under $R_j$.\\
b) For $j \in \llbracket 1, p \rrbracket$, let $s_j$ be the endomorphism of $E_0$ induced by $R_j$. Show that $(s_1, \ldots, s_p)$ is an H-system of $E_0$.\\
c) Deduce that $p(2m) \leqslant p(m) + 2$.