Integral Group Rings With All Central Units Trivial: Solvable Groups

Maheshwary, Sugandha

doi:10.1007/s13226-018-0260-0

Cited by 10 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This class of groups has recently again been extensively studied by G.K. Bakshi, S. Maheshwary and I.B.S. Passi [BMP17,Mah16]. We adopt their notation for these groups.…”

Section: Introductionmentioning

confidence: 99%

Integral group rings of solvable groups with trivial central units

Bächle

2017

Forum Mathematicum

View full text Add to dashboard Cite

The integral group ring $\mathbb{Z} G$ of a group $G$ has only trivial central units, if the only central units of $\mathbb{Z} G$ are $\pm z$ for $z$ in the center of $G$. We show that the order of a finite solvable group $G$ with this property, can only have $2$, $3$, $5$ and $7$ as prime divisors, 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.Comment: [v4]: 13 pages. Final version. To appear in "Forum Mathematicum" (already available online

show abstract

“…This class of groups has recently again been extensively studied by G.K. Bakshi, S. Maheshwary and I.B.S. Passi [BMP17,Mah16]. We adopt their notation for these groups.…”

Section: Introductionmentioning

confidence: 99%

Integral group rings of solvable groups with trivial central units

Bächle

2017

Forum Mathematicum

View full text Add to dashboard Cite

show abstract

“…Consequently, it follows that if V has finite abelianization, then the center of V consists only of trivial units and in this case, G is known as a cut groupcentral units trivial [BMP17]. This class of groups has been intensively studied lately [BMP17,Mah18,Bäc18,BCJM18,Tre19,BBM20]. From yet another perspective, understanding V /V ′ can also be regarded as a first important step towards understanding the lower central series of V , an important object for which very few results are available.…”

Section: Introductionmentioning

confidence: 99%

Abelianization of the unit group of an integral group ring

Bächle,

Maheshwary,

Margolis

2020

Preprint

Self Cite

View full text Add to dashboard Cite

For a finite group G and U := U(ZG), the group of units of the integral group ring of G, we study the implications of the structure of G on the abelianization U/U ′ of U . We pose questions on the connections between the exponent of G/G ′ and the exponent of U/U ′ as well as between the ranks of the torsion-free parts of Z(U ), the center of U , and U/U ′ . We show that the units originating from known generic constructions of units in ZG are well-behaved under the projection from U to U/U ′ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ + 18].The following variations of (2) appear natural for V : (R1) Is rank V /V ′ = rank Z(V )? (R2) Assume Z(V ) is finite. Is V /V ′ also finite?The above questions address the free subgroup of V /V ′ , while the ones listed next try to get hold of its torsion elements. Denote by exp H, the exponent of a group H, i.e., the greatest common divisor of the orders of torsion elements of H.

show abstract

“…[Seh93], Corollary 1.7). Further, a group G such that all central units in ZG are trivial, is termed as cut-group; this class of groups is currently a topic of active research ([BMP17], [Mah18], [Bäc18], [BCJM18], [BMP19], [Tre19]). In this article, we quite often use the characterization of abelian cut-groups, namely, an abelian group G is a cut-group, if and only if its exponent divides 4 or 6.…”

Section: Introductionmentioning

confidence: 99%