Lets $G$ is finite abelian group (such that for any $x\in G$ $x+x=0$, i.e. $G=\mathbb{Z}_{2}^{\oplus k}$ for some $k\in\mathbb{N}$) and $(\cdot,\cdot):G\times G\to \mathbb{Z}_{2}$ is symmetrical bilinear form.
Know that: $(a, m)=0,$ $(a, p)=1,$ $(b, m)=1,$ $(b, p)=0.$
Is it true that $(a, b) = 1?$
Thanks.