Hello, this is my first post.
I was wondering how, if we have a finite group and a field and an -module , how we would show there is no -submodule complementary to ? Obviously a proof by contradiction but how would I start?
then since are submodules, we'll have which is nonsense. hence which means that as an element of is non-zero. Q.E.D.
: note that is an ideal of i.e. it's both left and right sumodule. so if we assume that is a right submodule and then