Strong generating set

In abstract algebra, especially in the area of group theory, a strong generating set of a permutation group is a generating set that clearly exhibits the permutation structure as described by a stabilizer chain. A stabilizer chain is a sequence of subgroups, each containing the next and each stabilizing one more point.

Let $G\leq S_{n}$ be a group of permutations of the set $\{1,2,\ldots ,n\}.$ Let

$B=(\beta _{1},\beta _{2},\ldots ,\beta _{r})$

be a sequence of distinct integers, $\beta _{i}\in \{1,2,\ldots ,n\},$ such that the pointwise stabilizer of $B$ is trivial (i.e., let $B$ be a base for $G$). Define

$B_{i}=(\beta _{1},\beta _{2},\ldots ,\beta _{i}),\,$

and define $G^{(i)}$ to be the pointwise stabilizer of $B_{i}$. A strong generating set (SGS) for G relative to the base $B$ is a set

$S\subseteq G$

such that

$\langle S\cap G^{(i)}\rangle =G^{(i)}$

for each $i$ such that $1\leq i\leq r$.

The base and the SGS are said to be non-redundant if

$G^{(i)}\neq G^{(j)}$

for $i\neq j$.

A base and strong generating set (BSGS) for a group can be computed using the Schreier–Sims algorithm.

References

• A. Seress, Permutation Group Algorithms, Cambridge University Press, 2002.