= Solution
The flag makes every <composition factor> one-dimensional, described by a <character> $\lambda:H\to F$. The <Jordan–Hölder theorem for modules> for finite-length modules says that the multiset of simple composition factors is independent of the chosen composition series.
Here is a construction valid over every $F$. For $h\in H$, every <generalized eigenspace> of $\rho(h)$ is an $H$-<submodule>. Indeed $\operatorname{ad}(h)$ is nilpotent on $H$, say of exponent $c$, and
$$
(\rho(h)-aI)^N\rho(x)
=\sum_{j=0}^N\binom Nj\rho((\operatorname{ad}h)^jx)(\rho(h)-aI)^{N-j}.
$$
If $v$ is killed by $(\rho(h)-aI)^{N_0}$, take $N\ge N_0+c$. Terms with $j\ge c$ vanish by adjoint nilpotence, and all other terms kill $v$. This proves invariance of the <generalized eigenspace>.
The flag also makes the <characteristic polynomial> of every $\rho(h)$ split over $F$. Whenever two composition characters differ, choose an $h$ on which they differ and split the <module> into the distinct generalized eigenspaces of $\rho(h)$. Repeat within each summand until all its composition factors have the same <character>. Every nontrivial split lowers dimensions, so the process terminates. Finally combine summands carrying the same <character>. Jordan–Hölder ensures that the resulting distinct blocks have distinct factor types, giving
$$
\boxed{V=\bigoplus_\lambda W_\lambda,\qquad
\text{all composition factors of }W_\lambda\text{ have character }\lambda.}
$$
This recursive method does not assume that a single element separates every <character>, which could fail over a finite <field>.
Back to article page
