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

:$B\; =\; (\backslash beta\_1,\; \backslash beta\_2,\; \backslash ldots,\; \backslash beta\_r)$

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

:$B\_i\; =\; (\backslash beta\_1,\; \backslash beta\_2,\; \backslash ldots,\; \backslash beta\_i),\backslash ,$

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

:$S\; \backslash subseteq\; G$

such that

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

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

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

:$G^\{(i)\}\; \backslash neq\; G^\{(j)\}$

for $i\; \backslash neq\; j$.

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