TY - JOUR ID - 1396 TI - On the converse of Baer's theorem for generalizations of groups with trivial Frattini subgroups JO - Algebraic Structures and Their Applications JA - AS LA - en SN - 2382-9761 AU - Taghavi, Yasaman AU - Kayvanfar, Saeed AU - Chakaneh, Marzieh AD - Department of Pure Mathematics, Ferdowsi University of Mashhad, P. O. Box 1159, Mashhad 91775, Iran Y1 - 2019 PY - 2019 VL - 6 IS - 1 SP - 139 EP - 148 KW - Baer's theorem KW - Frattini subgroup KW - Upper and lower central series DO - 10.22034/as.2019.1396 N2 - 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. UR - https://as.yazd.ac.ir/article_1396.html L1 - https://as.yazd.ac.ir/article_1396_9ec887c2f69ccc83a253fc6414df4ebe.pdf ER -