## Central units of integral group rings

HTML articles powered by AMS MathViewer

- by Eric Jespers, Gabriela Olteanu, Ángel del Río and Inneke Van Gelder PDF
- Proc. Amer. Math. Soc.
**142**(2014), 2193-2209 Request permission

## Abstract:

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\mathbb {Z} G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $\mathcal {Z} (\mathcal {U} (\mathbb {Z} G))$ of the unit group $\mathcal {U} (\mathbb {Z} G)$ in case $G$ is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in $\mathcal {Z}(\mathcal {U}(\mathbb {Z} G))$ for all finite strongly monomial groups $G$. We call these units generalized Bass units. Finally, we show that the commutator group $\mathcal {U}(\mathbb {Z} G)/\mathcal {U}(\mathbb {Z} G)’$ and $\mathcal {Z}(\mathcal {U}(\mathbb {Z} G))$ have the same rank if $G$ is a finite group such that $\mathbb {Q} G$ has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of $\mathbb {Q}$. This allows us to prove that in this case the natural images of the Bass units of $\mathbb {Z} G$ generate a subgroup of finite index in $\mathcal {U}(\mathbb {Z} G)/\mathcal {U}(\mathbb {Z} G)’$.## References

- H. Bass,
*$K$-theory and stable algebra*, Inst. Hautes Études Sci. Publ. Math.**22**(1964), 5–60. MR**174604** - Hyman Bass,
*The Dirichlet unit theorem, induced characters, and Whitehead groups of finite groups*, Topology**4**(1965), 391–410. MR**193120**, DOI 10.1016/0040-9383(66)90036-X - Armand Borel and Harish-Chandra,
*Arithmetic subgroups of algebraic groups*, Ann. of Math. (2)**75**(1962), 485–535. MR**147566**, DOI 10.2307/1970210 - Charles W. Curtis and Irving Reiner,
*Representation theory of finite groups and associative algebras*, Pure and Applied Mathematics, Vol. XI, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR**0144979** - Raul Antonio Ferraz,
*Simple components and central units in group algebras*, J. Algebra**279**(2004), no. 1, 191–203. MR**2078394**, DOI 10.1016/j.jalgebra.2004.05.005 - Raul Antonio Ferraz and Juan Jacobo Simón-Pınero,
*Central units in metacyclic integral group rings*, Comm. Algebra**36**(2008), no. 10, 3708–3722. MR**2458401**, DOI 10.1080/00927870802158028 - J. Z. Gonçalves and D. S. Passman,
*Linear groups and group rings*, J. Algebra**295**(2006), no. 1, 94–118. MR**2188853**, DOI 10.1016/j.jalgebra.2005.02.009 - Eric Jespers, Ángel del Río, and Inneke Van Gelder,
*Writing units of integral group rings of finite abelian groups as a product of Bass units*, Math. Comp.**83**(2014), no. 285, 461–473. MR**3120600**, DOI 10.1090/S0025-5718-2013-02718-4 - Eric Jespers and Guilherme Leal,
*Generators of large subgroups of the unit group of integral group rings*, Manuscripta Math.**78**(1993), no. 3, 303–315. MR**1206159**, DOI 10.1007/BF02599315 - Eric Jespers and M. M. Parmenter,
*Construction of central units in integral group rings of finite groups*, Proc. Amer. Math. Soc.**140**(2012), no. 1, 99–107. MR**2833521**, DOI 10.1090/S0002-9939-2011-10968-7 - E. Jespers, M. M. Parmenter, and S. K. Sehgal,
*Central units of integral group rings of nilpotent groups*, Proc. Amer. Math. Soc.**124**(1996), no. 4, 1007–1012. MR**1328353**, DOI 10.1090/S0002-9939-96-03398-9 - Bernhard Liehl,
*On the group $\textrm {SL}_{2}$ over orders of arithmetic type*, J. Reine Angew. Math.**323**(1981), 153–171. MR**611449**, DOI 10.1515/crll.1981.323.153 - Aurora Olivieri, Ángel del Río, and Juan Jacobo Simón,
*On monomial characters and central idempotents of rational group algebras*, Comm. Algebra**32**(2004), no. 4, 1531–1550. MR**2100373**, DOI 10.1081/AGB-120028797 - Aurora Olivieri, Á. del Río, and Juan Jacobo Simón,
*The group of automorphisms of the rational group algebra of a finite metacyclic group*, Comm. Algebra**34**(2006), no. 10, 3543–3567. MR**2262368**, DOI 10.1080/00927870600796136 - Donald S. Passman,
*Infinite crossed products*, Pure and Applied Mathematics, vol. 135, Academic Press, Inc., Boston, MA, 1989. MR**979094** - César Polcino Milies and Sudarshan K. Sehgal,
*An introduction to group rings*, Algebra and Applications, vol. 1, Kluwer Academic Publishers, Dordrecht, 2002. MR**1896125**, DOI 10.1007/978-94-010-0405-3 - I. Reiner,
*Maximal orders*, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR**0393100** - Jürgen Ritter and Sudarshan K. Sehgal,
*Generators of subgroups of $U(\mathbf Z G)$*, Representation theory, group rings, and coding theory, Contemp. Math., vol. 93, Amer. Math. Soc., Providence, RI, 1989, pp. 331–347. MR**1003362**, DOI 10.1090/conm/093/1003362 - Jürgen Ritter and Sudarshan K. Sehgal,
*Construction of units in group rings of monomial and symmetric groups*, J. Algebra**142**(1991), no. 2, 511–526. MR**1127078**, DOI 10.1016/0021-8693(91)90322-Y - Jürgen Ritter and Sudarshan K. Sehgal,
*Trivial units in $RG$*, Math. Proc. R. Ir. Acad.**105A**(2005), no. 1, 25–39. MR**2138721**, DOI 10.3318/PRIA.2005.105.1.25 - 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** - K. Shoda,
*Über die monomialen darstellungen einer endlichen gruppe*, Proc. Phys.-math. Soc. Jap.**15**(1933), no. 3, 249–257. - Carl Ludwig Siegel,
*Discontinuous groups*, Ann. of Math. (2)**44**(1943), 674–689. MR**9959**, DOI 10.2307/1969104 - L. N. Vaseršteĭn,
*The group $SL_{2}$ over Dedekind rings of arithmetic type*, Mat. Sb. (N.S.)**89(131)**(1972), 313–322, 351 (Russian). MR**0435293** - L. N. Vaseršteĭn,
*Structure of the classical arithmetic groups of rank greater than $1$*, Mat. Sb. (N.S.)**91(133)**(1973), 445–470, 472 (Russian). MR**0349864** - T. N. Venkataramana,
*On systems of generators of arithmetic subgroups of higher rank groups*, Pacific J. Math.**166**(1994), no. 1, 193–212. MR**1306038**

## Additional Information

**Eric Jespers**- Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
- MR Author ID: 94560
- Email: efjesper@vub.ac.be
**Gabriela Olteanu**- Affiliation: Department of Statistics-Forecasts-Mathematics, Babeş-Bolyai University, Strada T. Mihali 58-60, 400591 Cluj-Napoca, Romania
- Email: gabriela.olteanu@econ.ubbcluj.ro
**Ángel del Río**- Affiliation: Departamento de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain
- MR Author ID: 288713
- Email: adelrio@um.es
**Inneke Van Gelder**- Affiliation: Department of Mathematics, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
- Email: ivgelder@vub.ac.be
- Received by editor(s): March 1, 2012
- Received by editor(s) in revised form: July 6, 2012
- Published electronically: March 27, 2014
- Additional Notes: This research was partially supported by Ministerio de Ciencia y Tecnología of Spain and Fundación Séneca of Murcia, the Research Foundation Flanders (FWO - Vlaanderen), Onderzoeksraad Vrije Universiteit Brussel and by the grant PN-II-RU-TE-2009-1 project ID_303 financed by the Romanian Ministry of National Education, CNCS-VEFISCDI
- Communicated by: Pham Huu Tiep
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**142**(2014), 2193-2209 - MSC (2010): Primary 16S34, 16U60, 16U70
- DOI: https://doi.org/10.1090/S0002-9939-2014-11958-7
- MathSciNet review: 3195747