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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| Format: | Article |
| Language: | English |
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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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| author | Mohammad K. Azarian |
| author_facet | Mohammad K. Azarian |
| author_sort | Mohammad K. Azarian |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-d65166d02aba44f5a39660cb2a8d4c06 |
| institution | OA Journals |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2001-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-d65166d02aba44f5a39660cb2a8d4c062025-08-20T02:19:40ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252001-01-0126211712110.1155/S0161171201005397Near Frattini subgroups of residually finite generalized free products of groupsMohammad K. Azarian0Department of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville 47722, IN, USALet 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.http://dx.doi.org/10.1155/S0161171201005397 |
| spellingShingle | Mohammad K. Azarian Near Frattini subgroups of residually finite generalized free products of groups International Journal of Mathematics and Mathematical Sciences |
| title | Near Frattini subgroups of residually finite generalized free products of groups |
| title_full | Near Frattini subgroups of residually finite generalized free products of groups |
| title_fullStr | Near Frattini subgroups of residually finite generalized free products of groups |
| title_full_unstemmed | Near Frattini subgroups of residually finite generalized free products of groups |
| title_short | Near Frattini subgroups of residually finite generalized free products of groups |
| title_sort | near frattini subgroups of residually finite generalized free products of groups |
| url | http://dx.doi.org/10.1155/S0161171201005397 |
| work_keys_str_mv | AT mohammadkazarian nearfrattinisubgroupsofresiduallyfinitegeneralizedfreeproductsofgroups |