Home Integral group rings of solvable groups with trivial central units
Article
Licensed
Unlicensed Requires Authentication

Integral group rings of solvable groups with trivial central units

  • Andreas Bächle EMAIL logo
Published/Copyright: October 27, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 4

Abstract

The integral group ring ℤ⁢G of a group G has only trivial central units if the only central units of ℤ⁢G are ±z for z in the center of G. We show that the order of a finite solvable group G with this property can only be divisible by the primes 2, 3, 5 and 7, by linking this to inverse semi-rational groups and extending one result on this class of groups. We also classify the Frobenius groups whose integral group rings have only trivial central units.

Keywords: Integral group ring; central units; inverse semi-rational; solvable groups; prime spectrum; Frobenius groups
MSC 2010: 16U60; 20C15; 20E45

Communicated by Manfred Droste

Funding statement: The author is a postdoctoral researcher of the FWO (Research Foundation Flanders).

Acknowledgements

The author is grateful to Silvio Dolfi for an interesting communication on inverse semi-rational groups and would like to thank Eric Jespers for stimulating discussions.

References

[1] S. Alavi, A. Daneshkhah and M. Darafsheh, On semi-rational Frobenius groups, J. Algebra Appl. 15 (2016), no. 2, Article ID 1650033. 10.1142/S021949881650033XSearch in Google Scholar

[2] G. Bakshi, S. Maheshwary and I. Passi, Integral group rings with all central units trivial, J. Pure Appl. Algebra 221 (2017), no. 8, 1955–1965. 10.1016/j.jpaa.2016.10.017Search in Google Scholar

[3] R. Brown and D. Harrison, Abelian Frobenius kernels and modules over number rings, J. Pure Appl. Algebra 126 (1998), no. 1–3, 51–86. 10.1016/S0022-4049(96)00144-2Search in Google Scholar

[4] D. Chillag and S. Dolfi, Semi-rational solvable groups, J. Group Theory 13 (2010), no. 4, 535–548. 10.1515/jgt.2010.004Search in Google Scholar

[5] M. R. Darafsheh and H. Sharifi, Frobenius ℚ-groups, Arch. Math. (Basel) 83 (2004), no. 2, 102–105. 10.1007/s00013-004-1020-4Search in Google Scholar

[6] R. Dark and C. Scoppola, On Camina groups of prime power order, J. Algebra 181 (1996), no. 3, 787–802. 10.1006/jabr.1996.0146Search in Google Scholar

[7] E. Farias e Soares, Big primes and character values for solvable groups, J. Algebra 100 (1986), no. 2, 305–324. 10.1016/0021-8693(86)90079-7Search in Google Scholar

[8] R. Gow, Groups whose characters are rational-valued, J. Algebra 40 (1976), no. 1, 280–299. 10.1016/0021-8693(76)90098-3Search in Google Scholar

[9] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss. 134, Springer, Berlin, 1967. 10.1007/978-3-642-64981-3Search in Google Scholar

[10] I. Isaacs and M. Lewis, Camina p-groups that are generalized Frobenius complements, Arch. Math. (Basel) 104 (2015), no. 5, 401–405. 10.1007/s00013-015-0755-4Search in Google Scholar

[11] E. Jespers and Á. del Río, Group Ring Groups. Volume 1: Orders and Generic Constructions of Units, De Gruyter, Berlin, 2016. 10.1515/9783110372946Search in Google Scholar

[12] S. Maheshwary, Integral group rings with all central units trivial: Solvable groups, preprint (2016), https://arxiv.org/abs/1612.08344v1; to appear in Indian J. Pure Appl. Math. 10.1007/s13226-018-0260-0Search in Google Scholar

[13] D. Passman, Permutation Groups, W. A. Benjamin, New York, 1968. Search in Google Scholar

[14] J. Ritter and S. Sehgal, Integral group rings with trivial central units, Proc. Amer. Math. Soc. 108 (1990), no. 2, 327–329. 10.1090/S0002-9939-1990-0994785-7Search in Google Scholar

[15] W. R. Scott, Group Theory, 2nd ed., Dover Publications, New York, 1987. Search in Google Scholar

[16] J. Tent, Quadratic rational solvable groups, J. Algebra 363 (2012), 73–82. 10.1016/j.jalgebra.2012.04.019Search in Google Scholar

[17] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.3, 2016, http://www.gap-system.org. Search in Google Scholar

Received: 2017-02-01
Revised: 2017-06-25
Published Online: 2017-10-27
Published in Print: 2018-07-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Distinct distances between points and lines in 𝔽q2
  3. A global perspective to connections on principal 2-bundles
  4. Integral group rings of solvable groups with trivial central units
  5. Lefschetz property and powers of linear forms in 𝕂[x,y,z]
  6. On quasi-convex null sequences in infinite cyclic groups
  7. On autocommutators and the autocommutator subgroup in infinite abelian groups
  8. Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
  9. Finite-dimensional representations of Leavitt path algebras
  10. Orlicz dual affine quermassintegrals
  11. Universal locally finite maximally homogeneous semigroups and inverse semigroups
  12. Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
  13. Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
  14. A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
  15. Equivariant Gauss sum of finite quadratic forms
  16. Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
  17. Solution of Brauer’s k⁢(B)-conjecture for π-blocks of π-separable groups
https://doi.org/10.1515/forum-2017-0021
Keywords for this article
Integral group ring; central units; inverse semi-rational; solvable groups; prime spectrum; Frobenius groups

Articles in the same Issue

  1. Frontmatter
  2. Distinct distances between points and lines in 𝔽q2
  3. A global perspective to connections on principal 2-bundles
  4. Integral group rings of solvable groups with trivial central units
  5. Lefschetz property and powers of linear forms in 𝕂[x,y,z]
  6. On quasi-convex null sequences in infinite cyclic groups
  7. On autocommutators and the autocommutator subgroup in infinite abelian groups
  8. Anticyclotomic p-ordinary Iwasawa theory of elliptic modular forms
  9. Finite-dimensional representations of Leavitt path algebras
  10. Orlicz dual affine quermassintegrals
  11. Universal locally finite maximally homogeneous semigroups and inverse semigroups
  12. Von Neumann algebras, L-algebras, Baer *-monoids, and Garside groups
  13. Boundedness and compactness of Hardy-type integral operators on Lorentz-type spaces
  14. A shifted convolution sum for \mathrm{GL}(3) × \mathrm{GL}(2)
  15. Equivariant Gauss sum of finite quadratic forms
  16. Right 2-Engel elements, central automorphisms and commuting automorphisms of Lie algebras
  17. Solution of Brauer’s k⁢(B)-conjecture for π-blocks of π-separable groups
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 5.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/forum-2017-0021/html
Scroll to top button