Let R be an associative and commutative ring with unity 1 and consider $k\in \mathbb{N}$ such that $1+1+\cdots +1=k$ is invertible. Let $U{T}_{\mathrm{\infty }}^{(k)}(R)$ be the group of upper triangular infinite matrices whose diagonal entries are kth roots of 1. We show that every element of the group $U{T}_{\mathrm{\infty }}(R)$ can be expressed as a product of 4k−6 commutators all depending on the powers of elements in $U{T}_{\mathrm{\infty }}^{(k)}(R)$ of order k. If R is the complex number field or the real number field we prove that, in $S{L}_n(R)$ and in the subgroup $S{L}_{VK}(\mathrm{\infty },R)$ 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.

设R是单位为 1 的结合交换环，并考虑$k\in \mathbb{否}$这样$\mathrm{1个}+\mathrm{1个}+\cdots +\mathrm{1个}=k$是可逆的。让$ü{吨}_{\mathrm{\infty }}^{(k)}(R)$是上三角无限矩阵的群，其对角元素是1 的第k个根。我们证明群中的每个元素$ü{吨}_{\mathrm{\infty }}(R)$可以表示为 4 k −6 个换向器的乘积，全部取决于元素的幂$ü{吨}_{\mathrm{\infty }}^{(k)}(R)$k阶。如果R是复数域或实数域，我们证明，在$\mathrm{小号}{\mathrm{大号}}_n(R)$并在子组中$\mathrm{小号}{\mathrm{大号}}_{V钾}(\mathrm{\infty },R)$对于R上的 Vershik–Kerov 群，这些群中的每个元素都可以分解为最多 4 k −6 个交换子的乘积，所有这些都取决于k阶的两个元素。