کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4585137 1630520 2013 47 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
On the unit conjecture for supersoluble group algebras
کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
On the unit conjecture for supersoluble group algebras
چکیده انگلیسی

We introduce structure theorems for the study of the unit conjecture for group algebras of torsion-free supersoluble groups. Motivated by work of P.M. Cohn we introduce the class of (X,Y,N)(X,Y,N)-group algebras KG  , and following D.S. Passman we define an induced length function L:KG→N∪{−∞}L:KG→N∪{−∞} using the fact that G has the infinite dihedral group as a homomorphic image. We develop splitting theorems   for (X,Y,N)(X,Y,N)-group algebras, and as an application show that if σ∈KGσ∈KG is a unit, then L(σ)=L(σ−1)L(σ)=L(σ−1). We extend our analysis of splittings to obtain a canonical reduced split-form   for all units in (X,Y,N)(X,Y,N)-group algebras. This leads to the study of group algebras of virtually abelian groups and their representations as subalgebras of suitable matrix rings, where we develop a determinant condition for units in such group algebras. We apply our results to the fours groupΓ=〈x,y|xy2x−1=y−2,yx2y−1=x−2〉 and show that over any field K, the group algebra KΓ has no non-trivial unit of small L-length. Using this, and the fact that L is equivariant under all KΓ-automorphisms obtained K-linearly from Γ  -automorphisms, we prove that no subset of the Promislow set P⊂ΓP⊂Γ is the support of a non-trivial unit in KΓ for any field K. In particular this settles a long-standing question and shows that the Promislow set is itself not the support of a unit in KΓ. We then give an introduction to the theory of consistent chains toward a preliminary analysis of units of higher L-length in KΓ. We conclude our work showing that units in torsion-free-supersoluble group algebras are bounded, in that the supports of units and their inverses are related through a property (U) and the induced length function L.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Algebra - Volume 394, 15 November 2013, Pages 310–356
نویسندگان
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