Split-complex number

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Revision as of 16:04, 12 December 2013 by en>BruceWMorlan (References and external links)
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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 GSn be a group of permutations of the set {1,2,,n}. Let

B=(β1,β2,,βr)

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

Bi=(β1,β2,,βi),

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

SG

such that

SG(i)=G(i)

for each i such that 1ir.

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

G(i)G(j)

for ij.

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.