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in article <1159599332.513347.176750@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
julien santini <julien_santini@xxxxxxxxx> wrote:
|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.
|
|===
|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 ?
|===
not if we're sufficiently competent. for example, take g to be the
permutations of h^3 where h is some finite non-abelian group (such as
the permutations of a 3-element set), take p to be 3, take a1 to be
the permutation that takes (h1,h2,h3) to (h2,h2^[-1]*h1*h2,h3), and
take a2 to be the permutation that takes (h1,h2,h3) to
(h1,h3,h3^[-1]*h2*h3).
--
jdolan@xxxxxxxxxxxx
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