Confusion regarding G-modules

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

Let be a reducible matrix representation with block form given by where are square matrices of the same size.

Let be a module for with submodule corresponding to . Consider the quotient vector space .

Show that is a -module with corresponding matrices . Furthermore, show that we have .

My confusion is in the definition of being a submodule (i.e. and the requirement that it is a module in its own right corresponding to matrices i.e. .

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