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finite number of finite order elements in a group

Source: RMO District 2005, 12th Grade, Problem 3

March 5, 2005
superior algebrasuperior algebra solved

Problem Statement

Let (G,⋅)(G,\cdot) be a group and let FF be the set of elements in the group GG of finite order. Prove that if FF is finite, then there exists a positive integer nn such that for all x∈Gx\in G and for all y∈Fy\in F, we have xny=yxn. x^n y = yx^n.
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