Abstract
We obtain some conditions for the virtual potency and virtual \( \pi \)-potency (where \( \pi \) is the set of primes) of some HNN-extensions and generalized free products of soluble minimax groups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Stebe P., “Conjugacy separability of certain free products with amalgamation,” Trans. Amer. Math. Soc., vol. 156, 119–129 (1971).
Allenby R.B.J.T., “The potency of cyclically pinched one-relator groups,” Arch. Math., vol. 36, no. 3, 204–210 (1981).
Hartley B., Lennox J.C., and Rhemtulla A.H., “Cyclically separated groups,” Bull. Aust. Math. Soc., vol. 26, no. 3, 355–384 (1982).
Wong P.C., Tang C.K., and Gan H.W., “Weak potency of fundamental groups of graphs of groups,” Bull. Malays. Math. Sci. Soc., vol. 33, no. 2, 243–251 (2010).
Azarov D.N., “On the weak \( \pi \)-potency of some groups and free products,” Sib. Math. J., vol. 61, no. 6, 953–962 (2020).
Baumslag G. and Solitar D., “Some two-generator one-relator non-Hopfian groups,” Bull. Amer. Math. Soc., vol. 68, no. 3, 199–201 (1962).
Hirsh K.A., “On infinite soluble groups,” J. Lond. Math. Soc., vol. 27, 81–85 (1952).
Lennox J. and Robinson D., The Theory of Infinite Soluble Groups, Clarendon, Oxford (2004).
Hsu T. and Wise D., “Ascending HNN-extensions of polycyclic groups are residually finite,” J. Pure Appl. Algebra, vol. 182, no. 1, 65–78 (2003).
Baumslag G., “On the residual finiteness of generalized free products of nilpotent groups,” Trans. Amer. Math. Soc., vol. 106, no. 2, 193–209 (1963).
Baumslag B. and Tretkoff M., “Residually finite HNN-extensions,” Comm. Algebra, vol. 6, no. 2, 179–194 (1978).
Lyndon R.C. and Schupp P.E., Combinatorial Group Theory, Springer, Berlin etc. (1977).
Rhemtulla A.H. and Shirvani M., “The residual finiteness of ascending HNN-extensions of certain soluble groups,” Illinois J. Math, vol. 47, no. 1, 477–484 (2003).
Azarov D.N., “On the residual finiteness of the HNN-extensions and generalized free products of finite rank groups,” Sib. Math. J., vol. 54, no. 6, 959–967 (2013).
Azarov D.N., “Approximability of generalized free products of groups with amalgamated normal subgroup by some classes of finite groups,” Sib. Math. J., vol. 56, no. 2, 206–216 (2015).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 6, pp. 1189–1203. https://doi.org/10.33048/smzh.2022.63.601
Rights and permissions
About this article
Cite this article
Azarov, D.N. On the Virtual Potency of Some Groups and Free Constructions. Sib Math J 63, 1023–1033 (2022). https://doi.org/10.1134/S0037446622060015
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446622060015
Keywords
- potent group
- residually finite group
- soluble minimax group
- HNN-extension
- Baumslag–Solitar group
- generalized free product