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.

References

1. Dixon, John D. (1996), Permutation Groups, Graduate Texts in Mathematics, 163, Springer, p. 76, ISBN 9780387945996.
2. Seress, Ákos (2003), Permutation Group Algorithms, Cambridge Tracts in Mathematics, 152, Cambridge University Press, pp. 1–2, ISBN 9780521661034, Sim's seminal idea was to introduce the notions of base and strong generating set.