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 groups and alternating groups have large bases (the symmetric group Sn has base size n − 1), and there are often specialized algorithms that deal with these cases.
