en>Johnpacklambert: /* External links */

2011-07-20T04:44:54Z

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Let <math>G</math> be a finite [[permutation group]] acting on a set <math>\Omega</math>. A sequence

:<math>B = [\beta_1,\beta_2,...,\beta_k]</math>

of ''k ''distinct elements of <math>\Omega</math> is a '''base''' for G if the only element of <math>G</math> which fixes every <math>\beta_i \in B</math> pointwise is the identity element of <math>G</math>.

We define the concept of a [[strong generating set]] relative to a base. Bases and strong generating sets are concepts of importance in [[computational group theory]]. A base and a strong generating set (together often called a BSGS) for a group can be obtained using the [[Schreier–Sims algorithm]].

It is often beneficial to deal with bases and strong generating sets as these may be easier to work with than the entire group. A group may have a small base compared to the set it acts on. In the "worst case", the [[symmetric group]]s and [[alternating group]]s have large bases (the symmetric group ''S''<sub>''n''</sub> has base size ''n'' − 1), and there are often specialized algorithms that deal with these cases.

{{algebra-stub}}

[[Category:Permutation groups]]
[[Category:Computational group theory]]

Chess'n Math Association - Revision history

en>Johnpacklambert: /* External links */