Expressing upper triangular matrices as products of commutators of finite order elements

Abstract

Listen

Translate article

Let R be an associative and commutative ring with unity 1 and consider such that is invertible. Let be the group of upper triangular infinite matrices whose diagonal entries are kth roots of 1. We show that every element of the group can be expressed as a product of 4k−6 commutators all depending on the powers of elements in of order k. If R is the complex number field or the real number field we prove that, in and in the subgroup of the Vershik–Kerov group over R, each element in these groups can be decomposed into a product of at most 4k−6 commutators, all depending on two elements of order k.