International Journal of Mathematics and Mathematical Sciences
Volume 26 (2001), Issue 2, Pages 117-121
doi:10.1155/S0161171201005397
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.
Abstract
Let G=A★HB be the generalized free product of the groups A and B with the amalgamated subgroup H. Also, let λ(G) and ψ(G) represent the lower near Frattini subgroup and the near Frattini subgroup of G, respectively. If G is finitely generated and residually finite, then we show that ψ(G)≤H, provided H satisfies a nontrivial identical relation. Also, we prove that if G is residually finite, then λ(G)≤H, provided: (i) H satisfies a nontrivial identical relation and A,B possess proper subgroups A1,B1 of finite index containing H; (ii) neither A nor B lies in the variety generated by H; (iii) H<A1≤A and H<B1≤B, where A1 and B1 each satisfies a nontrivial identical relation; (iv) H is nilpotent.