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IMC 2020 Problem 7

Source: IMC 2020

July 28, 2020
IMCgroup theoryabstract algebrasuperior algebraIMC 2020

Problem Statement

Let GG be a group and n≥2n \ge 2 be an integer. Let H1,H2H_1, H_2 be 22 subgroups of GG that satisfy [G:H1]=[G:H2]=n and [G:(H1∩H2)]=n(n−1).[G: H_1] = [G: H_2] = n \text{ and } [G: (H_1 \cap H_2)] = n(n-1). Prove that H1,H2H_1, H_2 are conjugate in G.G.
Official definitions: [G:H][G:H] denotes the index of the subgroup of H,H, i.e. the number of distinct left cosets xHxH of HH in G.G. The subgroups H1,H2H_1, H_2 are conjugate if there exists g∈Gg \in G such that g−1H1g=H2.g^{-1} H_1 g = H_2.
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