Concept

Strong generating set

Summary
In abstract algebra, especially in the area of group theory, a strong generating set of a permutation group is a generating set that clearly exhibits the permutation structure as described by a stabilizer chain. A stabilizer chain is a sequence of subgroups, each containing the next and each stabilizing one more point. Let G \leq S_n be a group of permutations of the set { 1, 2, \ldots, n }. Let : B = (\beta_1, \beta_2, \ldots, \beta_r) be a sequence of distinct integers, \beta_i \in { 1, 2, \ldots, n } , such that the pointwise stabilizer of B is trivial (i.e., let B be a base for G ). Define : B_i = (\beta_1, \beta_2, \ldots, \beta_i),, and define G^{(i)} to be the pointwise stabilizer of B_i . A strong generating set (SGS) for G relative to the base B is a set : S \subseteq G such that : \
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