Profinite groups with restricted centralizers of $$\pi $$-elements

A group G is said to have restricted centralizers if for each g in the centralizer $$C_G(g)$$ either finite or has index G. Shalev showed that a profinite with virtually abelian. Given set of primes $$\pi $$ , we take interest groups -elements. It shown such an open subgroup form $$P\times Q$$ where P abelian pro- and Q '$$ subgroup. This significantly strengthens result from our earlier paper.