Associative words in the symmetric group of degree three

Abstract

Let G be a group. An element \(w(x,y)\) of the absolutely free group on  free generators \(x,y\) is called an associative word in \(G\) if the equality \(w(w(g_1,g_2),g_3)=w(g_1,w(g_2,g_3))\) holds for all \(g_1,g_2 \in G\). In this paper we determine all associative words in the symmetric group  on three letters.