I'm having some trouble understanding my lecture notes.
I need to find a one-dimensional submodule $U$ of $\mathbb{C}^4$.
Is $u=1+x+x^2+x^3$ valid? My reasoning: because $ux = x + x^2 + x^3 + 1 = u \in U$. Would, say, $1+ix-x^2-ix^3$ be a different one?
I also have to express $\mathbb{C}^4$ as a direct sum of $\mathbb{C}G$-submodules ($G$ = <$x, y | x^4 = y^2=1, yxy=x^{-1}$>). Two of them have to be one-dimensional - does this mean the third must be two-dimensional? How do I do the direct sum of different dimensional things? I know this must seem like a very easy question but I really can't find a good explanation of 'direct sum' anywhere!
Thanks in advance!