Reflection classes whose cubes cover the alternating group

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Abstract

A reflection class (REC) over a finite set A is a conjugacy class of a reflection (permutation of order ≤ 2) of A. It was known that for no REC X, X2 = Alt(n) holds, and that for some RECs X, X4 = Alt(n) holds (n ≥ 5). Let i > 0, and let c(θ) denote the number of cycles of θ ε{lunate} S(n). Let Xi = {ψ ∈ S(n): ψ2 = 1, ψ has exactly i fixed points}. We prove that θ ε{lunate} Xi3 if and only if: (1) i ≡ n (mod 2); (2) The parity of Xi equals the parity of θ; and (3) i ≤ 1 3(n + 2 c(θ)). As a consequence, {X: X is a REC, X3 = Alt(n)} and {X: X is a REC, X3 = S(n) - Alt(n)} are determined.

Original languageEnglish
Pages (from-to)1-19
Number of pages19
JournalJournal of Combinatorial Theory. Series A
Volume21
Issue number1
DOIs
StatePublished - Jul 1976

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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