5,042 research outputs found
Free products in the unit group of the integral group ring of a finite group
Let be a finite group and let be a prime. We continue the search for
generic constructions of free products and free monoids in the unit group
of the integral group ring . For a
nilpotent group with a non-central element of order , explicit
generic constructions are given of two periodic units and in
such that ,
a free product of two cyclic groups of prime order. Moreover, if is
nilpotent of class and has order , then also concrete generators
for free products are constructed
(with ). As an application, for finite nilpotent groups, we
obtain earlier results of Marciniak-Sehgal and Gon{\c{c}}alves-Passman.
Further, for an arbitrary finite group we give generic constructions of
free monoids in that generate an infinite solvable
subgroup.Comment: 10 page
Generators of split extensions of Abelian groups by cyclic groups
Let be an -generator group with Abelian and
cyclic. We study the Nielsen equivalence classes and T-systems of generating
-tuples of . The subgroup can be turned into a finitely generated
faithful module over a suitable quotient of the integral group ring of .
When is infinite, we show that the Nielsen equivalence classes of the
generating -tuples of correspond bijectively to the orbits of unimodular
rows in under the action of a subgroup of . Making no
assumption on the cardinality of , we exhibit a complete invariant of
Nielsen equivalence in the case . As an application, we classify
Nielsen equivalence classes and T-systems of soluble Baumslag-Solitar groups,
lamplighter groups and split metacyclic groups.Comment: 36 pages, The former Theorem F.ii has been retracted because the
proof was wrong and couldn't be repaired. To appear in Groups, Geometry and
Dynamic
Stably free modules over virtually free groups
Let be the free group on generators and let be a finite
nilpotent group of non square-free order; we show that for each the
integral group ring has infinitely many stably free
modules of rank 1.Comment: 9 pages. The final publication is available at
http://www.springerlink.com doi:10.1007/s00013-012-0432-
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