On solvability of Engel equations in the group of triangular matrices over a field.
Let \(G\) be any group and \(k\geq 0\) an integer. The (left normed) \(k\)-Engel commutator \([x,{_ky}]\) for any two elements \(x,y\in G\) is defined inductively as follows: \([x,y]=[x,{_1y}]=x^{-1}y^{-1}xy\), \([x,{_ky}]=[[x,{_{k-1}y}],y]\) for all \(k>1\). In the paper under review the author denotes \([x,{_ky}]\) by \(e_k(x,y)\). Let \(K\) be a field of arbitrary characteristic. By \(UT_n(K)\), \(D_n(K)\) and \(T_n(K)\) we denote the subgroups of the general linear group \(GL_n(K)\), which are respectively the groups of upper unitriangular, diagonal and triangular matrices of size \(n\times n\) with all entries from the field \(K\). In the group \(UT_n(K)\), the author distinguishes subgroups \[ UT_n^m(K)=\{\mathbf 1_n+\sum_{i<j-m\leq n}a_{i,j}e_{i,j},\;a_{i,j}\in K\},\quad 0\leq m\leq n-1, \] where \(\mathbf 1_n\) denotes the unity matrix of size \(n\times n\) and \(e_{i,j}\) denotes the matrix with unity in the place \((i,j)\) and zeros elsewhere. Let \(\varphi_n\colon T_n(K)\hookrightarrow T_{n+1}(K)\) denote the natural embedding, i.e., \[ \varphi_n(A)=\left(\begin{smallmatrix} A &\underline 0^T\\ \underline 0&\mathbf 1\end{smallmatrix}\right)\quad\text{for all }A\in T_n(K), \] where \(\underline 0\) denotes a zero vector from \(K^n\). Then \(\varphi(UT_n(K))\subseteq UT_{n+1}(K)\) and \(\varphi_n(D_n(K))\subseteq D_{n+1}(K)\). Let \[ UT_\infty(K)=\varinjlim_n(UT_n(K),\varphi_n)\quad\text{and}\quad T_\infty(K)=\varinjlim_n(T_n(K),\varphi_n) \] denote finitary unitriangular and triangular matrices over the field \(K\). The main results of the paper under review are the following: Theorem 1. Let \(K\) be an arbitrary field. Then for every matrix \(C\) contained in \(UT^m_n(K)\) the Engel equation \(e_m(x,y)=C\) has a solution in \(UT_n(K)\). Theorem 2. Let \(K\) be a field containing at least \(n+1\) elements. Then for every matrix \(C\) contained in \(UT_n(K)\) the Engel equation \(e_k(x,y)=C\) has a solution in \(T_n(K)\), \(k\in\mathbb N\). Theorem 3. Let \(K\) be an arbitrary field. 1. For every matrix \(A\) contained in \(UT^m_\infty(K)\) the Engel equality \(e_m(x,y)=A\) has a solution in \(UT_\infty(K)\). 2. If \(K\) is infinite, then for every matrix \(A\) contained in \(UT_\infty(K)\) and any natural number \(k\) the Engel equation \(e_k(x,y)=A\) has a solution \(T_\infty(K)\). Some results are also proved for generalized Engel words. The author has also discussed the width problem of verbal subgroups of some special matrix groups. We refer the reader to the paper for details.
scientific article; zbMATH DE number 6138415