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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$w(H) = w(|H|)$$ \end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$v(G) = {\text{max\{ }}w(C(g))|g \in G\backslash Z(G)\} $$ \end{document}. In the present paper we present the complete description of groups with nontrivial center that satisfy the condition \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$v(G) = 4$$ \end{document}.