Base (group theory)

Let ${\displaystyle G}$ be a finite permutation group acting on a set ${\displaystyle \Omega }$. A sequence

${\displaystyle B=[\beta _{1},\beta _{2},...,\beta _{k}]}$

of k distinct elements of ${\displaystyle \Omega }$ is a base for G if the only element of ${\displaystyle G}$ which fixes every ${\displaystyle \beta _{i}\in B}$ pointwise is the identity element of ${\displaystyle G}$.[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.[2]

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.