A remark on a theorem of J. G. Thompson (Q1284722)

Let \(p\) be a prime. A finite group \(G\) has property \((p)\) if, for every \(g\in G-Z_G(G)\), \(p\) divides \(|G:C_G(g)|\). There exist non \(p\)-solvable groups with property \((p)\) (for example, \(\text{SL}(2,5)\) has property \((2)\)). In this note, the author shows that there do exist solvable groups with property \((p)\) of arbitrary large \(p\)-length. First, the solvable groups \(H\) of \(p\)-length \(2\) and property \((p)\) are constructed (for \(p=2\), the group \(\text{GL}(2,3)\) is used). Let \(G_k\) have property \((p)\) and \(p\)-length \(k\). Then \(G_{k+1}=G_k\text{ wr }H\), where \(H\) is considered as a transitive permutation group of degree \(|H|_p\), has property \((p)\) and \(p\)-length \(k+1\).