Groups with small centralizers (Q5961089)

For a natural \(n\), denote by \(w(n)\) the number of factors in a decomposition of \(n\) as a product of primes. For a finite group \(G\), let \(v(G)\) be the maximal value of \(w(|H|)\) where \(H\) runs over the set of all centralizers of non-central elements in \(G\). The authors give a complete description of finite groups with trivial center satisfying the condition \(v(G)=4\).