Base (group theory)

Let formula_1 be a finite permutation group acting on a set formula_2. A sequence 
of "k "distinct elements of formula_2 is a base for G if the only element of formula_1 which fixes every formula_6 pointwise is the identity element of formula_1.
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 "S""n" has base size "n" − 1), and there are often specialized algorithms that deal with these cases.