Editing 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 of a group|generating set]] that clearly exhibits the permutation structure as described by a '''stabilizer chain'''.  A stabilizer chain is a sequence of [[subgroup]]s, each containing the next and each stabilizing one more point.

Let <math>G \leq S_n</math> be a [[permutation group|group of permutations]] of the set <math>\{ 1, 2, \ldots, n \}.</math>  Let 

:<math> B = (\beta_1, \beta_2, \ldots, \beta_r) </math> 

be a sequence of distinct [[integers]], <math>\beta_i \in \{ 1, 2, \ldots, n \} ,</math> such that the [[Group action (mathematics)|pointwise stabilizer]] of <math> B </math> is trivial (i.e., let <math> B </math> be a [[Base (group theory)|base]] for <math> G </math>).  Define 

:<math> B_i = (\beta_1, \beta_2, \ldots, \beta_i),\, </math>

and define <math> G^{(i)} </math> to be the pointwise stabilizer of <math> B_i </math>. A '''strong generating set''' (SGS) for G relative to the base <math> B </math> is a [[Set (mathematics)|set]] 

:<math> S \subseteq G </math> 

such that 

:<math> \langle S \cap G^{(i)} \rangle = G^{(i)} </math> 

for each <math> i </math> such that <math> 1 \leq i \leq r </math>.

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

:<math> G^{(i)} \neq G^{(j)} </math> 

for <math> i \neq j </math>.

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

==References==
* A. Seress, ''Permutation Group Algorithms'',  Cambridge University Press, 2002.

{{DEFAULTSORT:Strong Generating Set}}
[[Category:Computational group theory]]
[[Category:Permutation groups]]