Approximate subgroup

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This article defines a property of subsets of groups
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Definition

Suppose d is a positive integer. A finite subset A of a group G is termed a d-approximate subgroup if it is a symmetric subset containing the identity element such that there exists a subset S of G of size at most d such that the product of subsets A^2 coincides with the product of subsets SA:

A^2 = SA

where A^2 = \{ a_1a_2 \mid a_1,a_2 \in A \} and SA = \{ sa \mid s \in S, a \in A \}

Note that any finite symmetric subset A containing the identity element is always a |A|-approximate subgroup, and A is a 1-approximate subgroup if and only if it is a subgroup. The minimum value of d for which A is a d-approximate subgroup describes how close A is to being a subgroup.

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