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(x^ky)^2=e if x^{n}=e,y^2=e,yxy=x^{-1}

Source: 1989 Greece MO Grade XII p4

September 7, 2024
group theoryalgebrasuperior algebra

Problem Statement

In a group GG, we have two elements x,yx,y such that xn=e,y2=e,yxy=x−1x^{n}=e,y^2=e,yxy=x^{-1}, n≥1n\ge 1. Prove that for any k∈Nk\in\mathbb{N} holds (xky)2=e(x^ky)^2=e.
Note : e=group's identity .
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