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.
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics