Document Type : Research Paper

**Authors**

Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran

10.29252/as.2019.1396

**Abstract**

In 2012, Guo and Gong proved that if $G$ is a finite nonabelian group with $\Phi(G)=1$, then $|G:Z(G)|<|G'||U(G)|$, in which $U(G)$ is the nilpotent residual of $G$. We show that the assumption of finiteness of the group can be omitted. Moreover, we investigate converse of Schur and Baer's theorems for groups that can be seen as generalizations of groups with trivial Frattini subgroups and state some properties of $n$-isoclinism families of these groups.

**Keywords**

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Winter and Spring 2019

Pages 139-148