Involutions of the automorphism group of a completely decomposable finite-rank abelian group (Q6546250)

If \(B\) is a completely decomposable torsion-free abelian group of finite rank written as a direct sum \(B = B_1\oplus B_2\oplus\dots\oplus B_s\) of its homogeneous components, then under a suitable ordering of the \(B_i\)'s, the automorphism group \(\Aut B\) can be identified with the group of all matrices of the form \(A=\left(\begin{array}{cccc} A_{11} & A_{12} & \ldots & A_{1 s} \\\N0 & A_{22} & \ldots & A_{2 s} \\\N\vdots & \vdots & \ddots & \vdots \\\N0 & 0 & \ldots & A_{s s} \end{array}\right)\) with \(A_{ij}\in\Hom(B_j, B_i)\) when \(1\le i<j\le s\) and \(A_{ii}\in\Aut B_i\) for \(1\le i\le s\). Motivated by the question of the definability of completely decomposable torsion-free abelian groups of finite rank by their automorphism groups, the authors study some properties of commuting involutions in \(\Aut B\). Let \(\mathop{\mathrm{Diag}}B\) stand for the subgroup of \(\Aut B\) consisting of all matrices with \(A_{ij}=0\) when \(1\le i<j\le s\) and let \(\Sigma\) denote the group of all matrices \(A\) such that \(A_{ij}\in\mathbb{Q}\otimes\Hom(B_j, B_i)\) when \(1\le i<j\le s\) and \(A_{ii}=\varepsilon_i\), the trivial automorphism of \(B_i\) for \(1\le i\le s\). For \(1\le k\le s\), let \(D_k\in\mathop{\mathrm{Diag}}B\) be the matrix with \[(D_k)_{ii}=\begin{cases}-\varepsilon_i&\text{if }i\le k\\\N\varepsilon_i&\text{if }i>k \end{cases}.\] The authors prove that if \(J_1, J_2, \dots, J_s\in\Aut B\) are commuting involutions such that for every \(k=1,\dots, s\), the set \(\{U^{-1}J_kUJ_k \mid U\in\Aut B\}\) is closed under multiplication and \(J_kD_k\in\Sigma\), then there exists a matrix \(T\in\Sigma\) such that \(T^{-1}J_kT = D_k\) for every \(k=1,\dots,s\) and \(\mathop{\mathrm{Diag}}B\subset T^{-1}(\Aut B)T\) (Theorem 3). Thus, commuting involutions satisfying the conditions of Theorem 3 are simultaneously conjugate to diagonal involutions while not in \(\Aut B\) but in a slightly larger group.