& &
              & & \bullet \ar[dl] \ar[dr] &  \\
         &    \bullet &  & \bullet & & \bullet & \bullet  \hspace{1 mm}
              \bullet \hspace{1 mm} \bullet  &  \bullet & & \bullet
        }$$

The reader may decide to make a pleasant check that the above diagrams are indeed representations of $V_{4}$. For each $V_{4}$-module $M$, one can define the dual $V_{4}$-module $M^{*}$, where $M^{*}$ is the dual $k$-vector space of $M$, and a group element $\sigma$ of $V_{4}$ acts on $f$ in $M$ via the rule $\sigma f(m) = f(\sigma^{-1} m)$. Then, as a fun exercise we ask the reader to verify that the diagrams in 3(a) and 3(b) are dual to each other. This will help the reader to get acquainted with some of the diagrammatic methods that will appear later on. We now begin by proving the easy part of the theorem.

\[lemma0\] All representations of $V_4$ that appear in the above theorem are indecomposable and pair-wise non-isomorphic.

Item (1) is not isomorphic to the rest because it is the only module that contains a non-zero element $x$ such that $(ab)x \ne 0$. Modules in item (2) are even dimensional and those in (3) are odd dimensional and hence there 