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• W1854352062 abstract "We present a Las Vegas algorithm which, for a given black-box group known to be isomorphic to a symmetric or alternating group, produces an explicit isomorphism with the standard permutation representation of the group. This algorithm has applications in computations with matrix groups and permutation groups. In this paper, we handle the case when the degree <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the standard permutation representation is part of the input. In a sequel, we shall treat the case when the value of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not known in advance. As an important ingredient in the theoretical basis for the algorithm, we prove the following result about the orders of elements of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper S Subscript n> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: the conditional probability that a random element <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma element-of upper S Subscript n> <mml:semantics> <mml:mrow> <mml:mi>σ<!-- σ --></mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma in S_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-cycle, given that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=sigma Superscript n Baseline equals 1> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>σ<!-- σ --></mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>sigma ^n=1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, is at least <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=1 slash 10> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>1/10</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
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• W1854352062 date "2003-01-14" @default.
• W1854352062 modified "2023-10-18" @default.
• W1854352062 title "A black-box group algorithm for recognizing finite symmetric and alternating groups, I" @default.
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• W1854352062 doi "https://doi.org/10.1090/s0002-9947-03-03040-x" @default.
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