Profinite groups with soluble centralisers

Abstract: We show that a profinite group, in which the centralisers of non-trivial elements are metabelian, is either virtually pro-$p$ or virtually soluble of derived length at most 4. We furthermore show that a prosoluble group, in which the centralisers of non-trivial elements are soluble of bounded derived length, is either soluble or virtually pro-$p$.