On Some Sets of Group Functions.

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