IMC 2020 · Problem 7

Let $G$ be a group and $n \ge 2$ be an integer. Let $H_1$ and $H_2$ be two subgroups of $G$ that satisfy

$$[G : H_1] = [G : H_2] = n \quad \text{and} \quad [G : (H_1 \cap H_2)] = n(n - 1).$$

Prove that $H_1$ and $H_2$ are conjugate in $G$.

(Here $[G : H]$ denotes the index of the subgroup $H$, i.e. the number of distinct left cosets $xH$ of $H$ in $G$. The subgroups $H_1$ and $H_2$ are conjugate if there exists an element $g \in G$ such that $g^{-1}H_1g = H_2$.)