On symmetric words in the symmetric group of degree three - Plonka, Ernest (MATHEMATICA SCANDINAVICA, 2006)
A word $w(x_{1},x_{2},…,x_{n})$ from absolutely free group $F_{n}$ is called symmetric $n$-word in a group $G$, if the equality $w(g_{1},g_{2},…,g_{n})=w(g_{σ 1},g_{σ 2},…,g_{σ n})$ holds for all $g_{1},g_{2},…,g_{n}\in G$ and all permutations $σ\in S_{n}$. The set $S^{(n)}(G)$ 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.