The product of two reflection classes of the symmetric group

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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.

Original languageEnglish
Pages (from-to)63-77
Number of pages15
JournalDiscrete Mathematics
Issue number1
StatePublished - 1976
Externally publishedYes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics


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