International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 2, Pages 117-121
Near Frattini subgroups of residually finite generalized free products of groups
Department of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville 47722, IN, USA
Received 3 May 1999; Revised 9 June 2000
Copyright © 2001 Mohammad K. Azarian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be the generalized free product of the groups and with the amalgamated subgroup . Also, let and represent the lower near Frattini subgroup and the near Frattini subgroup of , respectively. If is finitely generated and residually finite, then we show that , provided satisfies a nontrivial identical relation. Also, we prove that if is residually finite, then , provided: (i) satisfies a nontrivial identical relation and possess proper subgroups of finite index containing ; (ii) neither nor lies in the variety generated by ; (iii) and , where and each satisfies a nontrivial identical relation; (iv) is nilpotent.