Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 4 / 3 / b / Solution

For generators $g,h$ of $P$, write $x=g-1$, $y=h-1$. ThenFor each $\lambda \in k$, define a two-dimensional module $M_{\lambda }$ on basis $v,w$ by $xv=w$, $xw=0$, $yv=\lambda w$, $yw=0$. Every product of two of these operators is zero, so the defining relations hold even when $p=2$.

The action of $x$ is one nonzero nilpotent Jordan block. A decomposition of $M_{\lambda }$ would decompose its restriction to $k[x]/(x^p)$, which is already indecomposable. Thus $M_{\lambda }$ is indecomposable. An isomorphism $F:M_{\lambda }\to M_{\mu }$ commutes with $x$ and $y$, and consequentlySince $F$ is invertible and $x\ne 0$, $\lambda =\mu$. The infinite field supplies infinitely many parameters, proving $kP$ has infinite representation type. This is the parameter family for an elementary abelian p-group.