Alternate modules are subsymplectic

In this paper, an alternate module $(A,\varphi )$ is a finite abelian group $A$ with a $\mathbb{Z}$-bilinear application $\phi:A\times A\rightarrow $\mathbb{Q}$/$\mathbb{Z}$$ which is alternate (i.e. zero on the diagonal). We shall prove that any alternate module is subsymplectic, i.e. if $(A,\varphi )$ has a Lagrangian of cardinal $n$ then there exists an abelian group $B$ of order $n$ such that $(A,\varphi )$ is a submodule of the standard symplectic module $B\times B^*$.