## 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| = א_{0} and X is a conjugacy class in S_{A} of infinite support, then X^{4} = S_{A}, and asked whether 3 can replace 4. Theorem. For |A| ≥ א_{0}, X^{3} ≠ S_{A}, where X is the conjugacy class of reflections with no fixed point.

Original language | English |
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Pages (from-to) | 63-77 |

Number of pages | 15 |

Journal | Discrete Mathematics |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - 1976 |

Externally published | Yes |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics