On the regularity of Sylow subgroups of full linear groups over residue rings (Q1905335)

Let \(p\) be a prime, \(m\) and \(n\) positive integers and \(P(m,n)\) the group of all \(n\) by \(n\) matrices \((a_{ij})\) over the integers modulo \(p^m\) with \(a_{ii} \equiv 1\) and \(a_{ij} \equiv 0\) modulo \(p\), for all \(i\) and all \(j < i\). Solving a problem posed by the reviewer in the Kourovka Notebook, the author proves the following. The \(p\)-group \(P(m,n)\) is regular if and only if one of the following hold. (i) \(n = 1\). (ii) \(p > 7\) and either \(m = 1\) and \(2 \leq n \leq p\); or \(m = 2\) and \(2 \leq n \leq 5\); or \(m = 3\) and \(n = 2\). (iii) \(p = 7\) and either \(m = 1\) and \(2 \leq n \leq 7\); or \(m = 2\) and \(n = 2, 3\); or \(m = 3\) and \(n = 2\). (iv) \(p = 5\) and either \(m = 1\) and \(2 \leq n \leq 5\); or \(m = 2, 3\) and \(n = 2\). (v) \(p = 3\), \(m = 1\) and \(n = 2, 3\). (vi) \(p = 2\), \(m = 1\) and \(n = 2\).