International Mathematics Competition
for University Students
2020

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

Hint: Express \(\displaystyle H_1H_2\) both as the disjoint union of left cosets with respect to \(\displaystyle H_2\) and and as the disjoint union of right cosets with respect to \(\displaystyle H_1\).

  Solution