Coprime numbers and intersection of subgroups We know that the intersection of two subgroups is {identity element} if their orders are coprime (proof via Lagrange's Theorem). But can we say: if the intersection of two finite subgroups is {identity element}, then their orders are coprime? I think yes, I tried to prove the contrary, but I stucked. Could you please give me a hint? 

No, it's not true. Consider for example $\mathbb Z_2\times\mathbb Z_2$. There are three proper nontrivial subgroups of the same order that all intersect trivially pairwise.
