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x+y=z in Fp * subgroup, 6|order

Source: VJIMC 2014 2.2

May 21, 2021
group theoryabstract algebra

Problem Statement

Let pp be a prime number and let AA be a subgroup of the multiplicative group Fp∗\mathbb F^*_p of the finite field Fp\mathbb F_p with pp elements. Prove that if the order of AA is a multiple of 66, then there exist x,y,z∈Ax,y,z\in A satisfying x+y=zx+y=z.
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