On the converse of Baer's theorem for generalizations of groups with trivial Frattini subgroups
نویسندگان
1 Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
2 Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
3 Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran
doi
10.22034/as.2019.1396چکیده
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.
