Groups

["Julien Santini"]

Given a group G with order n, p an odd prime, and a_1,...,a_p p
elements of G such that a_1a_2=a_2a_3=...=a_pa_1, show that n is
divisible by 2p.

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I could show that n is divisible by 2 (all a_i have same even order),
(a_i)^2=(a_k)^2 for all i,k, a_ia_k=a_(i+1)a_(k+1) for all i,k. The
last assertion gives that the set {a_i,a_j; p>=i,j>=1} is a subgroup of
G with order p, provided that we can prove that (a_i)^2=1 for at least
one i. Can we show this ?
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Thanks.