Strong generating set

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 ${\displaystyle G\leq S_{n}}$ be a group of permutations of the set ${\displaystyle \{1,2,\ldots ,n\}.}$ Let

be a sequence of distinct integers, ${\displaystyle \beta _{i}\in \{1,2,\ldots ,n\},}$ such that the pointwise stabilizer of ${\displaystyle B}$ is trivial (i.e., let ${\displaystyle B}$ be a base for ${\displaystyle G}$). Define

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