Sims-reduced generating set

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This article defines a property that can be evaluated for a generating set of a permutation group: a group equipped with an embedding in a symmetric group on an ordered set

Definition

Let G be a group acting on a set S = \{ 1, 2, 3, \ldots, n \} of size n (equivalent G is equipped with an embedding in the symmetric group on n elements). A Sims-reduced generating set is a generating set for G where for any elements i < j, there is at most one elements of G which fixes all elements less than i and sends i to j.

Any generating set for G can be trimmed down to a Sims-reduced generating set using the Sims filter.

Stronger properties