The product of two reflection classes of the symmetric group

Gadi Moran

doi:10.1016/0012-365X(76)90110-2

The product of two reflection classes of the symmetric group

Gadi Moran

Research output: Contribution to journal › Article › peer-review

Abstract

Necessary and sufficient conditions for a permutation to be a product of two reflections (permutations of order ≤ 2) from a given pair of conjugacy classes are presented. Various corolaries are derived. Examples: (1) Ore showed that a set A is infinite iff every permutation as a commutator in S_A. Theorem. A set A is uncountable iff every permutation is a commutator of two reflections. (2) Bertram sharpened a theorem of Ulam and Schreier, showing that if |A| = א₀ and X is a conjugacy class in S_A of infinite support, then X⁴ = S_A, and asked whether 3 can replace 4. Theorem. For |A| ≥ א₀, X³ ≠ S_A, where X is the conjugacy class of reflections with no fixed point.

Original language	English
Pages (from-to)	63-77
Number of pages	15
Journal	Discrete Mathematics
Volume	15
Issue number	1
DOIs	https://doi.org/10.1016/0012-365X(76)90110-2
State	Published - 1976
Externally published	Yes

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/0012-365X(76)90110-2

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title = "The product of two reflection classes of the symmetric group",

abstract = "Necessary and sufficient conditions for a permutation to be a product of two reflections (permutations of order ≤ 2) from a given pair of conjugacy classes are presented. Various corolaries are derived. Examples: (1) Ore showed that a set A is infinite iff every permutation as a commutator in SA. Theorem. A set A is uncountable iff every permutation is a commutator of two reflections. (2) Bertram sharpened a theorem of Ulam and Schreier, showing that if |A| = א0 and X is a conjugacy class in SA of infinite support, then X4 = SA, and asked whether 3 can replace 4. Theorem. For |A| ≥ א0, X3 ≠ SA, where X is the conjugacy class of reflections with no fixed point.",

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year = "1976",

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AB - Necessary and sufficient conditions for a permutation to be a product of two reflections (permutations of order ≤ 2) from a given pair of conjugacy classes are presented. Various corolaries are derived. Examples: (1) Ore showed that a set A is infinite iff every permutation as a commutator in SA. Theorem. A set A is uncountable iff every permutation is a commutator of two reflections. (2) Bertram sharpened a theorem of Ulam and Schreier, showing that if |A| = א0 and X is a conjugacy class in SA of infinite support, then X4 = SA, and asked whether 3 can replace 4. Theorem. For |A| ≥ א0, X3 ≠ SA, where X is the conjugacy class of reflections with no fixed point.

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The product of two reflection classes of the symmetric group

Abstract

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