Originally Posted by

**AtticusRyan** The following question illustrates the confusion I have (from *The Symmetric Group* by Bruce Sagan):

Let $\displaystyle X$ be a reducible matrix representation with block form given by $\displaystyle \left(\begin{array}{cc}A(g)&B(g)\\0&C(g)\end{array }\right)$ where $\displaystyle A,B,C$ are square matrices of the same size.

Let $\displaystyle V$ be a module for $\displaystyle X$ with submodule $\displaystyle W$ corresponding to $\displaystyle A$. Consider the quotient vector space $\displaystyle V/W=\{\mathbf{v}+W\mid \mathbf{v}\in W\}$ (v in V?).

Show that $\displaystyle V/W$ is a $\displaystyle G$-module with corresponding matrices $\displaystyle C(g)$. Furthermore, show that we have $\displaystyle V\cong W\oplus (V/W)$.

My confusion is in the definition of $\displaystyle W$ being a submodule (i.e. $\displaystyle gw\in W$ and the requirement that it is a module in its own right corresponding to matrices $\displaystyle A(g)$ i.e. $\displaystyle gw=A(g)w$.

How does one do this question? I have trouble getting started and confusion with all the definitions. Any help would be greatly appreciated!

Many thanks,

Atticus