Near Frattini subgroups of residually finite generalized free products of groups
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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2001-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171201005397 |
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| Summary: | 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. |
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| ISSN: | 0161-1712 1687-0425 |