Reflection classes whose cubes cover the alternating group

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, X 2 = Alt( n) holds, and that for some RECs X, X 4 = Alt( n) holds ( n ⩾ 5). Let i > 0, and let c( θ) denote the number of cycles of θ ϵ S( n). Let X i = { ψ ∈ S( n): ψ 2 = 1, ψ has exactly i fixed points}. We prove that θ ϵ X i 3 if and only if: (1) i ≡ n (mod 2); (2) The parity of X i equals the parity of θ; and (3) i ⩽ 1 3 (n + 2 c(θ)) . As a consequence, { X: X is a REC, X 3 = Alt( n)} and { X: X is a REC, X 3 = S( n) − Alt( n)} are determined.