Strong Generating Set
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 permutation group. Let B = (beta_1, beta_2, ldots, beta_r) be a sequence of distinct integers, beta_i in { 1, 2, ldots, n } , such t...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 permutation group. 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 (ie: 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 Bi. A strong generating set (SGS) for G relative to the base B is a set S subset G such that langle S cap G^{(i)} rangle = G^{(i)} for each 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.
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