On symmetric words in the symmetric group of degree three

Abstract

A word $w(x_{1},x_{2},\ldots,x_{n})$ from absolutely free group $F_{n}$ is called symmetric $n$-word in a group $G$, if the equality $w(g_{1},g_{2},\ldots,g_{n})=w(g_{\sigma 1},g_{\sigma 2},\ldots,g_{\sigma n})$ holds for all $g_{1},g_{2},\ldots,g_{n}\in G$ and all permutations $\sigma\in S_{n}$. The set $\mathbf{S}^{(n)}(G)$ of all symmetric $n$-words is a subgroup of $F_{n}$. In this paper the groups of all symmetric $2$-words and $3$-words for the symmetric group of degree 3 are determined.