| Let $f\colon {\Bbb G}_{n,k} \longrightarrow {\Bbb G}_{m,l}$ be any continuous map between two{\itdistinct} complex (resp. quaternionic) Grassmann manifolds of the same dimension. We show that the degree of $f$ is zero provided $n,m$ are sufficiently large and $l \geq 2$. If the degree of $f$ is $\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also, we prove that the image under $f^*$ of elements of a set of algebra generators of $H^*({\Bbb G}_{m,l}; \Bbb{Q})$ is determined up to a sign, $\pm$, if the degree of $f$ is non-zero. \\
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