Let

G

be a finite

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...

acting on a set

\Omega

. A sequence :

B = beta_1,\beta_2,...,\beta_k /math> 

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

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 The Schreier–Sims algorithm is an algorithm in computational group theory, named after the mathematicians Otto Schreier and Charles Sims. This algorithm can find the order of a finite permutation group, test membership (is a given permutation ...

.. 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.

References

{{algebra-stub Permutation groups Computational group theory