Let $A$ be an elementary abelian p-group and let $\mathbf a\in(a_1,\dots,a_r)\in A^r$. Let $f_\mathbf a:(\mathbb Z/p\mathbb Z)\to A$ be the unique homomorphism satisfying \(f_\mathbf a(1,0,\dot,0) = a_1, \dots, f_\mathbf a(0,\dots, 1) = a_r.\) If the function $f_\mathbf a$ is injective, then the $a_i$ are linearly independent.
Wikidata ID: Q27670