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Top-stable and layer-stable degenerations and hom-order

S. O. Smalø, A. Valenta (2007)

Colloquium Mathematicae

Using geometrical methods, Huisgen-Zimmermann showed that if M is a module with simple top, then M has no proper degeneration M < d e g N such that t M / t + 1 M t N / t + 1 N for all t. Given a module M with square-free top and a projective cover P, she showed that d i m k H o m ( M , M ) = d i m k H o m ( P , M ) if and only if M has no proper degeneration M < d e g N where M/M ≃ N/N. We prove here these results in a more general form, for hom-order instead of degeneration-order, and we prove them algebraically. The results of Huisgen-Zimmermann follow as consequences from our results....

Torsion matrices over commutative integral group rings.

Gregory T. Lee, Sudarshan K. Sehgal (2000)

Publicacions Matemàtiques

Let ZA be the integral group ring of a finite abelian group A, and n a positive integer greater than 5. We provide conditions on n and A under which every torsion matrix U, with identity augmentation, in GLn(ZA) is conjugate in GLn(QA) to a diagonal matrix with group elements on the diagonal. When A is infinite, we show that under similar conditions, U has a group trace and is stably conjugate to such a diagonal matrix.

Torsion units for some almost simple groups

Joe Gildea (2016)

Czechoslovak Mathematical Journal

We investigate the Zassenhaus conjecture regarding rational conjugacy of torsion units in integral group rings for certain automorphism groups of simple groups. Recently, many new restrictions on partial augmentations for torsion units of integral group rings have improved the effectiveness of the Luther-Passi method for verifying the Zassenhaus conjecture for certain groups. We prove that the Zassenhaus conjecture is true for the automorphism group of the simple group PSL ( 2 , 11 ) . Additionally we prove that...

Torsion units in group rings.

Vikas Bist (1992)

Publicacions Matemàtiques

Let U(RG) be the unit group of the group ring RG. In this paper we study group rings RG whose support elements of every torsion unit are torsion, where R is either the ring of integers Z or a field K.

Totally indefinite Euclidean quaternion fields

Jean-Paul Cerri, Jérôme Chaubert, Pierre Lezowski (2014)

Acta Arithmetica

We study the Euclidean property for totally indefinite quaternion fields. In particular, we establish a complete list of norm-Euclidean such fields over imaginary quadratic number fields. This enables us to exhibit an example which gives a negative answer to a question asked by Eichler. The proofs are both theoretical and algorithmic.

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