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| Paper IPM / M / 18017 |
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| Abstract: | |
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The aim of this paper is to consider Waring's problem for ${\SL}_n(D)$, where $D$ is a division ring with center $F$ such that $\dim_F D\leq 4$. We show that each element of ${\SL}_n(D)$ is a product of at most three square elements. As an application, let $FG$ be a group algebra of a locally finite group $G$ over a field $F$ of characteristic $p\neq 2$. We show that if either $p>2$, or $F$ is algebraically closed, or $F$ is real-closed, or $G$ is locally nilpotent, then every element in the derived subgroup $(FG)'$ is a product of at most three squares.
In this paper, we also discuss about decompositions of elements into products of torsion elements by showing, for instance, that if a field $F$ contains at least $n$ distinct torsion elements, then every element in $\mathrm{SL}_n(F)$ is a product of at most two torsion elements.
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