Base (group theory)

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Let $G$ be a finite permutation group acting on a set $Ω$ . A sequence

B = [β 1,β 2,...,β k]

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

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 S_n has base size n − 1), and there are often specialized algorithms that deal with these cases.