Result: Powers of Cycle-Classes in Symmetric Groups: Powers of cycle-classes in symmetric groups

The authors extend a preceding result obtained by the first author on the alternating group \(A_n\). They prove that each permutation of \(A_n\) (\(n\geq 1\)) is a product of three \(l\)-cycles in \(S_n\) if and only if \(l\) is odd and either \(\lceil n/2\rceil\leq l\leq n\) or \(n=7\) and \(l=3\). They also prove that each permutation of \(A_n\) (\(n\geq 2\)) is a product of four \(l\)-cycles in \(S_n\) if and only if \(\lceil 3n/8\rceil\leq l\leq n\) if \(n\not\equiv 1\pmod 8\) or \(\lfloor 3n/8\rfloor\leq l\leq n\) if \(n\equiv 1\pmod 8\) or \(n=6\) and \(l=2\). They conjecture that, for fixed \(k\) and \(n\), if \(f\) is the minimal integer such that each permutation of \(A_n\) is a product of \(k\) \(f\)-cycles in \(S_n\) then \(|f-3n/k|\leq 2+\varepsilon(k)\) where \(\varepsilon(k)=0\) for \(k\) even and \(1\) for \(k\) odd.