Factorization of matrices into products of squares or torsions and its applications to group algebras

The aim of this paper is to consider Waring's problem for SLn(D), where D is a division ring with center F such that dimF D <= 4. We show that each element of SLn(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 not equal 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 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 SLn(F) is a product of at most two torsion elements.