On Some Sets of Group Functions

Let $G$ be a group, let $A$ be an Abelian group, and let $n$ be an integer such that $n\ge-1$. In the paper, the sets $\Phi_n(G,A)$ of functions from $G$ into $A$ of degree not greater than $n$ are studied. In essence, these sets were introduced by Logachev, Sal'nikov, and Yashchenko. We describe all cases in which any function from $G$ into $A$ is of bounded (or not necessarily bounded) finite degree. Moreover, it is shown that if $G$ is finite, then the study of the set $\Phi_n(G,A)$ is reduced to that of the set $\Phi_n(G/O^p(G),A_p)$ for primes $p$ dividing $|G/G'|$. Here $O^p(G)$ stands for the $p$-coradical of the group $G$, $A_p$ for the $p$-component of $A$, and $G'$ for the commutator subgroup of $G$.