Geodesic
On Some Sets of Group Functions
Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 3-11 
M. I. Anokhin. On Some Sets of Group Functions. Matematičeskie zametki, Tome 74 (2003) no. 1, pp. 3-11. http://geodesic.mathdoc.fr/item/MZM_2003_74_1_a0/
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     title = {On {Some} {Sets} of {Group} {Functions}},
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     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_1_a0/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $G$.

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