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 ≤ S n be a group of permutations of the set { 1 , 2 , … , n } . Let

be a sequence of distinct integers, β i ∈ { 1 , 2 , … , n } , such that the pointwise stabilizer of B is trivial (i.e., let B be a base for G ). Define

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

such that

for each i such that 1 ≤ i ≤ r .

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

for i ≠ j .

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