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.
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Shumyatsky et al. (2024) studied this question.
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