transformation $$a + \lambda b : V_1 \otimes_k k[\lambda] {\ensuremath{\rightarrow}}V_2 \otimes_k k[\lambda]$$ is singular, i.e, $det( a + \lambda b) = 0$.

2.  Dually, $\Omega^{-l} (k)$ is a quotient of $M$ for some positive integer $l$ if and only if the transformation $$a^* + \lambda b^* : V_2^* \otimes_k k[\lambda] {\ensuremath{\rightarrow}}V_1^* \otimes_k k[\lambda]$$ is singular.

The next lemma is very crucial to the classification. To the best of our knowledge, nowhere in the literature is this lemma stated explicitly, although it is secretly hidden in Benson’s proof of the classification [@ben-1]. We use Auslander-Reiten sequences to give a transparent proof of this lemma in the last section.

\[lemma2\] Let $M$ be a projective-free $V_4$-module. Then we have the following.

1.  If $l$ is the smallest positive integer such that $\Omega^l(k)$ is isomorphic to a submodule of $M$, then $\Omega^l(k)$ is a summand of $M$.

2.  Dually, if $l$ is the smallest positive integer such that $\Omega^{-l}(k)$ is isomorphic to a quotient module of $M$, then $\Omega^{-l}(k)$ is a $G$-summand of $M$.

[@johnson-1]\[lemma3\] For all integers $n$, $\Omega^n (k)$ is isomorphic to the dual representation $\Omega^{-n} (k) ^*$. Furthermore,

1.  If $n$ is positive, then $\Omega^n (k)$ is a $2n+1$ dimensional 