Parameter family for an elementary abelian p-group
= Parameter family for an elementary abelian p-group
For $P=C_p\times C_p$, put $x=g-1$, $y=h-1$. On a basis $v,w$, prescribe $xv=w$, $xw=0$, $yv=\lambda w$, $yw=0$. These are valid modules because every product of two radical operators is zero. Each is indecomposable, and an isomorphism preserves the equality $y=\lambda x$, hence preserves $\lambda$. An infinite field therefore gives infinitely many isomorphism classes.