Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RESIDUAL p-FINITENESS OF CERTAIN HNN EXTENSIONS OF FREE ABELIAN GROUPS OF FINITE RANK

  • Chiew Khiam Tang (Institute of Mathematical Sciences Faculty of Science, University of Malaya) ;
  • Peng Choon Wong (Institute of Mathematical Sciences Faculty of Science, University of Malaya)
  • Received : 2023.07.04
  • Accepted : 2023.12.28
  • Published : 2024.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let p be a prime. A group G is said to be residually p-finite if for each non-trivial element x of G, there exists a normal subgroup N of index a power of p in G such that x is not in N. In this note we shall prove that certain HNN extensions of free abelian groups of finite rank are residually p-finite. In addition some of these HNN extensions are subgroup separable. Characterisations for certain one-relator groups and similar groups including the Baumslag-Solitar groups to be residually p-finite are proved.

Keywords

References

  1. S. Andreadakis, E. Raptis, and D. Varsos, Residual finiteness and Hopficity of certain HNN extensions, Arch. Math. (Basel) 47 (1986), no. 1, 1-5. https://doi.org/10.1007/BF01202492 
  2. S. Andreadakis, E. Raptis, and D. Varsos, Extending isomorphisms to automorphisms, Arch. Math. (Basel) 53 (1989), no. 2, 121-125. https://doi.org/10.1007/BF01198560 
  3. G. Baumslag, Lectures on nilpotent groups, C.B.M.S. Regional Conference Series 2, Amer. Math. Soc., 1981. 
  4. G. Baumslag and D. Solitar, Some two-generator one-relator non-Hopfian groups, Bull. Amer. Math. Soc. 68 (1962), 199-201. https://doi.org/10.1090/S0002-9904-1962-10745-9 
  5. W. P. Choon and W. K. Bin, The residual finiteness of certain HNN extensions, Bull. Korean Math. Soc. 42 (2005), no. 3, 555-561. https://doi.org/10.4134/BKMS.2005.42.3.555 
  6. K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math. Soc. (3) 7 (1957), 29-62. https://doi.org/10.1112/plms/s3-7.1.29 
  7. M. Hall, Coset representations in free groups, Trans. Amer. Math. Soc. 67 (1949), 421-432. https://doi.org/10.2307/1990483 
  8. G. Higman, Amalgams of p-groups, J. Algebra 1 (1964), 301-305. https://doi.org/10.1016/0021-8693(64)90025-0 
  9. K. Iwasawa, Einige Satze uber freie Gruppen, Proc. Imp. Acad. Tokyo 19 (1943), 272-274. http://projecteuclid.org/euclid.pja/1195573488 
  10. A. Karrass, A. Pietrowski, and D. M. Solitar, Finite and infinite cyclic extensions of free groups, J. Austral. Math. Soc. 16 (1973), 458-466.  https://doi.org/10.1017/S1446788700015445
  11. G. Kim and J. McCarron, On amalgamated free products of residually p-finite groups, J. Algebra 162 (1993), no. 1, 1-11. https://doi.org/10.1006/jabr.1993.1237 
  12. G. Kim and J. McCarron, Some residually p-finite one relator groups, J. Algebra 169 (1994), no. 3, 817-826. https://doi.org/10.1006/jabr.1994.1310 
  13. G. Kim and C. Y. Tang, Polygonal products which are residually finite p-groups, in Group theory (Granville, OH, 1992), 275-287, World Sci. Publ., River Edge, NJ, 1993. 
  14. G. Kim and C. Y. Tang, On generalized free products of residually finite p-groups, J. Algebra 201 (1998), no. 1, 317-327. https://doi.org/10.1006/jabr.1997.7256 
  15. G. Kim and C. Y. Tang, Cyclic subgroup separability of HNN-extensions with cyclic associated subgroups, Canad. Math. Bull. 42 (1999), no. 3, 335-343. https://doi.org/10.4153/CMB-1999-039-4 
  16. J. McCarron, Residually nilpotent one-relator groups with nontrivial centre, Proc. Amer. Math. Soc. 124 (1996), no. 1, 1-5. https://doi.org/10.1090/S0002-9939-96-03148-6 
  17. E. Raptis and D. Varsos, Some residual properties of certain HNN extensions, Bull. Soc. Math. Gr'ece (N.S.) 28 (1987), part A, 81-87. 
  18. E. Raptis and D. Varsos, Residual properties of HNN-extensions with base group an abelian group, J. Pure Appl. Algebra 59 (1989), no. 3, 285-290. https://doi.org/10.1016/0022-4049(89)90098-4 
  19. E. Raptis and D. Varsos, The residual nilpotence of HNN-extensions with base group a finite or a f.g. abelian group, J. Pure Appl. Algebra 76 (1991), no. 2, 167-178. https://doi.org/10.1016/0022-4049(91)90059-B
  20. N. S. Romanovski˘i, On the residual finiteness of free products with respect to subgroups, Izv. Akad. Nauk SSSR Ser. Mat. 33 (1969), 1324-1329. 
  21. P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17 (1978), no. 3, 555-565. https://doi.org/10.1112/jlms/s2-17.3.555 
  22. P. C. Wong, Subgroup separability of certain HNN extensions, Rocky Mountain J. Math. 23 (1993), no. 1, 391-394. https://doi.org/10.1216/rmjm/1181072631 
  23. P. C. Wong, Subgroup separability of certain HNN extensions of finitely generated abelian groups, Rocky Mountain J. Math. 27 (1997), no. 1, 359-365. https://doi.org/10.1216/rmjm/1181071967 
  24. P. C. Wong and C. K. Tang, Tree products of residually p-finite groups, Algebra Colloq. 2 (1995), no. 3, 209-212. 
  25. P. C. Wong and C. K. Tang, Free products of residually p-finite groups with commuting subgroups, Bull. Malaysian Math. Soc. (2) 19 (1996), no. 1, 25-28.