Abstract

It has been proved in J.A. Green (1956) [5] for every p  -group of order pⁿ, , where t(G)⩾0. In Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], and X. Zhou (1994) [14], the structure of G   has been characterized for t(G)=0,1,2,3by several authors. Also in A.R. Salemkar et al. (2007) [12], the structure of G  characterized when t(G)=4 and Z(G) is elementary abelian, but there are some missing points in classifying the structure of these groups. This paper is devoted to classify the structure of G   when t(G)=4 without any condition and with a short and quite different way to that of Ya.G. Berkovich (1991) [1], G. Ellis (1999) [4], A.R. Salemkar et al. (2007) [12], and X. Zhou (1994) [14].