Consider the group $T_p(n)$ of all non-singular upper triangular matrices with entries in $\mathbb{F}_p.$ Its commutator subgroup is $U_p(n)$ (all elements in $T_p(n)$ with $1$s on the main diagonal). The question is: is every element in $U_p$ a commutator of matrices in $T_p?$

## 1 Answer

Too long for a comment and only a partial answer for $n>2$. **A note on commutators in the group of infinite triangular matrices over a ring**, by Agnieszka Bier and Waldemar Holubowski, *Linear and Multilinear Algebra* **63** (2015) no. 11, 2301-2310; seems relevant. It includes results on finite matrices, despite the title.

Let $R$ be an associative ring with $1$, and let ${U}(R)$ be its group of invertible elements. $T(n,R)$ denotes the group of upper triangular $n\times n$ matrices; $UT(n,R)$ the set of upper unitriangular matrices ($1$s in the diagonal), $UT(n,m,R)$ the subgroup containing exactly all those matrices which have zero entries on the first $m$ super diagonals; and the basic commutators $c_k(x_1,\ldots,x_k)$ are defined inductively by $c_1(x_1)=x_1$, $c_{i+1}(x_1,\ldots,x_{i+1}) = [c_i(x_1,\ldots,x_i),x_{i+1}]$; let $\gamma_k(G)=G$ be the $k$th term of the lower central series of $G$. Finally, the Engel words $e_k(x,y)$ are given by $e_2(x,y)=[x,y]$, $e_{m+1}(x,y) = [e_m(x,y),y]$.

The authors prove:

Theorem 1.5Let $R$ be an associative ring with $1$ such that $U(R)$ is commutative. Then:

- $\gamma_k(UT(n,R))=UT(n,k,R)$ and every element of $\gamma_k(UT(n,R))$ is a value of the basic commutator $c_k$.
- Every element of $\gamma_k(UT(n,R))$ is a value of the Engel word $e_k$.
Moreover, if $1$ is a sum of two invertible elements, then

- $[T(n,R),T(n,R)] = UT(n,R)$ whenever $n\geq 2$.
- Every element of $\gamma_2(T(2,R)) = UT(2,R)$ is a commutator.
- Every element of $\gamma_2(T(n,R)) = UT(n,R)$ with $n>2$ is a product of at most two commutators.

In particular, for $R=\mathbb{F}_p$ with $p>2$, then the answer is "yes" for $n=2$ by point 4; but for $n>2$ they only show every element is a product of at most two commutators.

Perhaps this was already known for the special case of matrices over commutative rings/fields?

63(2015) no. 11, 2301-2310) gives that the answer is "yes" for $n=2$ and $p$ odd; and for $n>2$, every element is a product of at most two commutators of matrices in $T_p(n)$. $\endgroup$5more comments