In abstract algebra

Abstract algebra

Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, especially in the area of group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, a strong generating set of a permutation group

Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...

 is a generating set

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

 that clearly exhibits the permutation structure as described by a stabilizer chain. A stabilizer chain is a sequence of subgroup

Subgroup

In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

s, each containing the next and each stabilizing one more point.

Let be a group of permutations

Permutation group

In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...

 of the set Let


be a sequence of distinct integers, such that the pointwise stabilizer

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

 of is trivial (i.e., let be a base

Base (group theory)

Let G be a finite permutation group acting on a set \Omega. A sequenceB = [\beta_1,\beta_2,...,\beta_k]of k distinct elements of \Omega is a base for G if the only element of G which fixes every \beta_i \in B pointwise is the identity element of G....

 for ). Define


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


such that


for each such that .

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


for .

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