On elements of prime-power index in finite groups. (Q2268832)

Let \(G\) be a finite group and \(x\) an element in \(G\). The index \(\text{Ind}_G(x)\) of \(C_G(x)\) in \(G\) is an important quantity. In this paper, the authors prove that if \(\text{Ind}_{\langle a,b,x\rangle}(x)\) is a prime-power for any \(a,b\in G\), then \(\text{Ind}_G(x)\) is a prime-power.