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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 Hindawi Publishing Corporation. 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 G=AHB 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<A1A and H<B1B, where A1 and B1 each satisfies a nontrivial identical relation; (iv) H is nilpotent.