Generalizations of product-free subsets

Details Generalizations of product-free subsets Generalizations of product-free subsets

2008-04-04

arxiv.org/abs/0804.0687v1

Mathematics

Group Theory

8 pages; from conference "Communicating Mathematics" in honor of Joe Gallian (Duluth, 2007); related to 0708.2295

Scientific paper

For any group G of order n, a subset A of G is said to be product-free if there is no solution of the equation ab=c with a,b,c in A. Previous results of Gowers showed that the size of any product-free subset of G is at most n/d^(1/3), where d is the smallest dimension of a nontrivial representation of G. However, this upper bound does not match the best lower bound. We will generalize the upper bound to the case of product-poor subsets A, in which the equation ab=c is allowed to have a few solutions with a,b,c in A. We prove that the upper bound for the size of product-poor subsets matches the best lower bound in many families of groups. We will also generalize the concept of product-free to the case in which we have many subsets of a group, and different constraints about products of the elements in the subsets.

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