Let G be a finite permutation group acting on a set Ω. A sequence In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself); the relationship is often written as (G,M). ...

B = [β₁,β₂,...,β_k]

of k distinct elements of Ω is a base for G if the only element of G which fixes every pointwise is the identity element of G.

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. Let be a permutation group. ... In mathematics, computational group theory is the study of groups by means of computers. ... The Schreier-Sims algorithm is an efficient method of computing a strong generating set (SGS) of a permutation group. ...

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. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics an alternating group is the group of even permutations of a finite set. ...