Symmetric words in groups".

W. Holubowski
(Sil. Tech. U., Poland)

An $n-$ary word $w$ is called an $n-$symmetric word in a group $G$ (or simply: a symmetric word) if (*) $w(g_1, \ldots, g_n)=w(g_{\sigma(1)}, \ldots, g_{\sigma(n)})$ for all $g_1, \ldots, g_n$ from $G$ and all permutations $\sigma$ from the symmetric group $G$.

Symmetric words in a given group $G$ are closely connected with fixed points of the automorphisms permuting generators in the corresponding relatively free group, and also with symmetric operations in universal algebras and symmetric identities.

The $n-$symmetric words in $G$ form a group $S^{(n)}(G)$.

I give a survey of results concerning generators of $S^{(n)}(G)$ in nilpotent, soluble and finite groups.