Functionally complete groups

A group $G$ is called functionally complete if for an arbitrary natural number $n$ every mapping $f:G^n\to G$ can be realized by a «polynomial» in at most $n$ variables over the group $G$. We know that a group $G$ is functionally complete if and only if it is either trivial or a finite simple non-Abelian group [Ref. Zh. Mat. 9A174 (1975)]. In this article the ldquodegreerdquo of a polynomial and the connected notions of $n$-functional completeness, $(n;k_1,\dots,k_n)$-functional completeness, and strong functional completeness are introduced. It is shown that for $n>1$ these notions and the notion of functional completeness are equivalent, and apart from all finite simple non-Abelian groups, only the trivial group and groups of second order are 1-functionally complete.