Constructing free products of cyclic subgroups inside the group of units of integral group rings
HTML articles powered by AMS MathViewer
- by Zbigniew Marciniak and Sudarshan K. Sehgal;
- Proc. Amer. Math. Soc. 151 (2023), 1487-1493
- DOI: https://doi.org/10.1090/proc/16249
- Published electronically: January 30, 2023
- HTML | PDF | Request permission
Abstract:
It has been proved in Janssens, Jespers, and Temmerman [Proc. Amer. Math. Soc. 145 (2017), pp. 2771–2783] that if $h$ is an element of prime order $p$ in a finite nilpotent group $G$ and $u=h+(h-1)g\widehat {h}\in \mathbb {Z}G$, $u\not \in G$, then $\langle u^*,u\rangle \approx C_p\ast C_p$. We offer a simple geometric approach to generalize this result.References
- V. Bovdi, Free subgroups in group rings, arXiv:1406.6771, 2014, a preprint
- Geoffrey Janssens, Eric Jespers, and Doryan Temmerman, Free products in the unit group of the integral group ring of a finite group, Proc. Amer. Math. Soc. 145 (2017), no. 7, 2771–2783. MR 3637929, DOI 10.1090/proc/13631
- J. Z. Gonçalves and D. S. Passman, Embedding free products in the unit group of an integral group ring, Arch. Math. (Basel) 82 (2004), no. 2, 97–102. MR 2047662, DOI 10.1007/s00013-003-4793-y
- Jairo Z. Gonçalves and Ángel Del Río, A survey on free subgroups in the group of units of group rings, J. Algebra Appl. 12 (2013), no. 6, 1350004, 28. MR 3063443, DOI 10.1142/S0219498813500047
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 89, Springer-Verlag, Berlin-New York, 1977. MR 577064
- Zbigniew S. Marciniak and Sudarshan K. Sehgal, Constructing free subgroups of integral group ring units, Proc. Amer. Math. Soc. 125 (1997), no. 4, 1005–1009. MR 1376998, DOI 10.1090/S0002-9939-97-03812-4
- Zbigniew S. Marciniak and Sudarshan K. Sehgal, Subnormal subgroups of group ring units, Proc. Amer. Math. Soc. 126 (1998), no. 2, 343–348. MR 1423318, DOI 10.1090/S0002-9939-98-04126-4
- Donald S. Passman, The algebraic structure of group rings, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. MR 470211
- D. S. Passman, Free products in linear groups, Proc. Amer. Math. Soc. 132 (2004), no. 1, 37–46. MR 2021246, DOI 10.1090/S0002-9939-03-07033-3
- Sudarshan K. Sehgal, Topics in group rings, Monographs and Textbooks in Pure and Applied Mathematics, vol. 50, Marcel Dekker, Inc., New York, 1978. MR 508515
- S. K. Sehgal, Units in integral group rings, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 69, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993. With an appendix by Al Weiss. MR 1242557
Bibliographic Information
- Zbigniew Marciniak
- Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02–097 Warsaw, Poland
- MR Author ID: 119635
- Email: zbimar@mimuw.edu.pl
- Sudarshan K. Sehgal
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, T6G 2G1, Canada
- MR Author ID: 158130
- Email: ss5@ualberta.ca
- Received by editor(s): February 11, 2022
- Received by editor(s) in revised form: September 2, 2022
- Published electronically: January 30, 2023
- Communicated by: Martin Liebeck
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1487-1493
- MSC (2020): Primary 20C05
- DOI: https://doi.org/10.1090/proc/16249
- MathSciNet review: 4550344