\(d\)-wise generation of prosolvable groups.
Given a topological group \(G\), and \(d\geq 2\), let \(\mu_d(G)\) denote the largest \(m\) for which there exists an \(m\)-tuple of elements of \(G\) such that any \(d\)-tuple extracted from it generates \(G\) (as a topological group). In the paper under review the authors obtain a lower bound for \(\mu_d(G)\) for \(G\) prosoluble. From this it follows that if \(G\) is \(d\)-generated, then \(\mu_d(G)-d\) tends to infinity as the smallest prime divisor of \(|G|\) tends to infinity. The authors call the attention to an intriguing problem in linear algebra that underlies this question. Let \(F\) be a finite field, \(t\geq d>0\) integers, and \(A_1,\dots,A_t\) matrices over \(F\). Suppose that for all \(1\leq i_1<\cdots<i_d\leq t\) the \(n\times nd\) matrix \((A_{i_1},\dots,A_{i_d})\) has maximum rank \(n\). Is it always possible to find matrices \(B_1,\dots,B_t\) of dimension \(n(d-1)\times n\) over \(F\) such that the square matrices obtained by putting \((A_{i_1},\dots,A_{i_d})\) on top of \((B_{i_1},\dots,B_{i_d})\) have non-zero determinant for all \(1\leq i_1<\cdots<i_d\leq t\)? In general, the answer is negative, but it would be interesting to prove positive results for certain values of the parameters. We refer to the paper for a detailed discussion.
scientific article; zbMATH DE number 6155223