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 $$G \leq S_n$$ 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


 * $$ \langle S \cap G^{(i)} \rangle = G^{(i)} $$

for each $$ i $$ such that $$ 1 \leq i \leq r $$.

The base and the SGS are said to be non-redundant if


 * $$ G^{(i)} \neq G^{(j)} $$

for $$ i \neq j $$.

A base and strong generating set (BSGS) for a group can be computed using the Schreier–Sims algorithm.