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Permutations destroying arithmetic progressions in finite cyclic groups

Peter Hegarty (Institutionen för matematiska vetenskaper, matematik) ; Anders Martinsson (Institutionen för matematiska vetenskaper, matematik)
The Electronic Journal of Combinatorics (1077-8926). Vol. 22 (2015), 4, p. Art. no. P4.39.

A permutation \pi of an abelian group G is said to destroy arithmetic progressions (APs) if, whenever (a,b,c) is a non-trivial 3-term AP in G, that is c-b=b-a and a,b,c are not all equal, then (\pi(a),\pi(b),\pi(c)) is not an AP. In a paper from 2004, the first author conjectured that such a permutation exists of Z/nZ, for all n except 2,3,5 and 7. Here we prove, as a special case of a more general result, that such a permutation exists for all n >= n_0, for some explcitly constructed number n_0 \approx 1.4 x 10^{14}. We also construct such a permutation of Z/pZ for all primes p > 3 such that p = 3 (mod 8).

Nyckelord: Permutation, arithmetic progression, finite cyclic group

CPL Pubid: 218523

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, matematik (2005-2016)