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?
Thanks,
John
suppose and let see that for all and thus for all we have for some and so now suppose that
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