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Quasirandom Groups

  • W. T. GOWERS (a1)

Babai and Sós have asked whether there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy = z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.

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[1] L. Babai and L. Rónyai (1990) Computing irreducible representations of finite groups Math. Comp. 55 705722.

[2] L. Babai and V. Sós (1985) Sidon sets in groups and induced subgraphs of Cayley graphs. Europ. J. Combin. 6 101114.

[3] B. Bollobás and V. Nikiforov (2004) Hermitian matrices and graphs: Singular values and discrepancy. Discrete Math. 285 1732.

[5] F. R. K. Chung and R. L. Graham (1992) Quasi-random subsets of ℤn. J. Combin. Theory Ser. A 61 6486.

[6] F. R. K. Chung , R. L. Graham and R. M. Wilson (1989) Quasi-random graphs. Combinatorica 9 345362.

[8] W. T. Gowers (2001) A new proof of Szemerédi's theorem. Geom. Funct. Anal. 11 465588.

[10] I. M. Isaacs (2006) Character Theory of Finite Groups, AMS Chelsea Publishing, Providence, RI (corrected reprint of 1976 original).

[11] K. S. Kedlaya (1997) Large product-free subsets of finite groups. J. Combin. Theory Ser. A 77 339343.

[12] K. S. Kedlaya (1998) Product-free subsets of groups. Amer. Math. Monthly 105 900906.

[14] A. Lubotzky , R. Phillips and P. Sarnak (1988) Ramanujan graphs. Combinatorica 8 261277.

[16] P. Sarnak and X. Xue (1991) Bounds for multiplicities of automorphic representations. Duke Math. J. 64 207227.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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