Two subgroups whose orders are greater than square root of the group order have no trivial intersection

Question

Two subgroups whose orders are greater than square root of the group order have no trivial intersection

I cannot come up with a critical idea.

Answer 1

Hint:

Recall that if $H$ and $K$ are finite subgroups of $G$, then

$$o(HK)=\frac{o(H)o(K)}{o(H\cap K)}$$

I am using $o(L)$ to denote the order of $L$.

You also know $HK$ may itself not be a subgroup, but certainly every element of $HK$ is an element of $G$.

Answer 2

Use $|H_1H_2| \cdot |H_1 \cap H_2| = |H_1||H_2]$ for $H_1,H_2$ subgroups of a group $G$.