# International Mathematics Competition for University Students 2020

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IMC 2022
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## IMC2020: Day 2, Problem 7

Problem 7. Let $\displaystyle G$ be a group and $\displaystyle n\ge2$ be an integer. Let $\displaystyle H_1$ and $\displaystyle H_2$ be two subgroups of $\displaystyle G$ that satisfy

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

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

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

Ilya Bogdanov and Alexander Matushkin, Moscow Institute of Physics and Technology

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