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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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