On Finite Groups with Restrictions on Centralizers

Denote by $w(n)$ the number of factors in a representation of a positive integer $n$ as a product of primes. If $H$ is a subgroup of a finite group $G$, then we set $w(H)=w(|H|)$ and $v(G)=\max \{w(C(g))\mid g\in G\setminus Z(G)\}$. In the present paper we present the complete description of groups with nontrivial center that satisfy the condition $v(G)=4$.