Quotient of semi-simple representation is semi-simple

Question

Prove that Quotient of semi-simple representation is semi-simple.

Take $V=\oplus_i V_i$ a semi-simple representation of finite dimension of a finite group.

For a fixed j, we have $V/V_j=\oplus_i (V_i+V_j)/V_j$ (Is that even true? I am inspired by R-modules here).

Then the question is whether $(V_i+V_j)/V_j$ is simple for each $i$?

Please let me know what you think of this approach, or if there is simpler way.

Thank you.

Answer 1

If $M$ is a submodule of $V=\bigoplus V_i$, where all the $V_i$'s are simple, then you can use the second isomorphism theorem to get $$V/M \simeq \bigoplus \frac{V_i}{V_i\cap M}.$$

But since $V_i$ is simple, $V_i\cap M$ is either $0$ or $V_i$. So each factor $V_i/(V_i\cap M)$ is either $0$ or $V_i$. Thus $$V/M \simeq \bigoplus_{j\in J} V_j$$ where $J$ is some index set. This is a sum of simples.

Edit: Linear representations of a group are just modules over its group algebra. So definitely use modules to inspire your thoughts on this subject. Many undergrads neglect this.