Exercise 4, § 7

Let E be a free A-module of dimension $n \geq 3$. Show that if $1 \leq p \leq n - 1$ the A-modules $\bigwedge^p E$ and $\bigwedge^{n-p} E$ are isomorphic; but if $2p \neq n$, there exists no isomorphism $\phi$ of $\bigwedge^p E$ onto $\bigwedge^{n-p} E$ such that for every automorphism $u$ of E, $\phi \circ (\bigwedge^p u) = (\bigwedge^{n-p} u) \circ \phi$ (if $(e_i)$ is a basis of E, consider the automorphism $u$ such that $u(e_i) = e_i + e_j, y(e_k) = e_k$ for $k \neq i, j$, and give i and j all possible values).