Hello I was reading Serre's book Linear Representations of finite groups and had a doubt while reading a proof of a proposition (Proposition 24, pg. 61):
Proposition 24. Let A be a normal subgroup of a group G, and let $\rho: \mathbf{G} \rightarrow \mathbf{G L}(\mathrm{V})$ be an irreducible representation of $\mathrm{G}$. Then:
(a) either there exists a subgroup $\mathrm{H}$ of $\mathrm{G}$, unequal to $\mathrm{G}$ and containing $\mathrm{A}$, and an irreducible representation $\sigma$ of $\mathrm{H}$ such that $\rho$ is induced by $\boldsymbol{\sigma}$;
(b) or else the restriction of $\rho$ to $\mathrm{A}$ is isotypic.
(A representation is said to be isotypic if it is a direct sum of isomorphic irreducible representations.)
The proof is as follows:
Let $V=\oplus V_{i}$ be the canonical decomposition of the representation $\rho$ (restricted to A) into a direct sum of isotypic representations (cf. 2.6). For $s \in G$ we see by "transport de structure" that $\rho(s)$ permutes the $V_{i}$; since $V$ is irreducible, $G$ permutes them transitively. Let $V_{i_{0}}$ be one of these; if $V_{i_{0}}$ is equal to $V$, we have case (b). Otherwise, let $H$ be the subgroup of $G$ consisting of those $s \in G$ such that $\rho(s) V_{i_{0}}=V_{i_{0}}$. We have $A \subset H$, $H \neq G$, and $\rho$ is induced by the natural representation $\sigma$ of $H$ in $V_{i_{0}}$, which is case (a).
I have two questions:
- Why does $\rho(s)$ permute the $V_i?$
- Why is $A\subseteq H?$
Some help with these questions would mean a lot!
Thank you so much!
Best Answer
If $Z \subset V$ is an irreducible $A$-submodule, then because $A$ is normal, $\rho(s) A$ is also an irreducible $A$-submodule. Hence $\rho(s) V_i$ must be one of the isotypic summands $V_j$.
This is simply because the $V_i$ are submodules for $A$.