Constructing free subgroups of integral group ring units

Marciniak, Zbigniew; Sehgal, Sudarshan K.

doi:10.1090/s0002-9939-97-03812-4

Cited by 41 publications

(9 citation statements)

References 5 publications

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“…A result of Tits, [13,Lemma 1.5.3] gives necessary conditions for a pair of diagonalizable elements of GL(2, C) to generate a free group. This problem is also addressed in several other papers (see [9], [10] and [11]).…”

Section: Introductionmentioning

confidence: 92%

Free groups in quaternion algebras

Juriaans

Filho

2013

Journal of Algebra

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In [8] we constructed pairs of units u, v in Z-orders of a quaternion algebra over Q( √ −d), d ≡ 7 (mod 8) positive and square free, such that u n , v n is free for some n ∈ N. Here we extend this result to any imaginary quadratic extension of Q, thus including matrix algebras. More precisely, we show that u n , v n is a free group for all n ≥ 1 and d > 2 and for d = 2 and all n ≥ 2. The units we use arise from Pell's and Gauss' equations. A criterion for a pair of homeomorphisms to generate a free semigroup is also established and used to prove that two certain units generate a free semigroup but that, in this case, the Ping-Pong Lemma can not be applied to show that the group they generate is free. * Mathematics Subject Classification Primary [16U 60, 20E05]; Secondary [16S34, 20M 05].

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Section: Introductionmentioning

confidence: 92%

Free groups in quaternion algebras

Juriaans

Filho

2013

Journal of Algebra

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“…In fact the bicyclic units β and γ of Theorem 1.2 can be explicitly constructed (see Corollary 4.3). The existence of a free pair formed by bicyclic units of different types was already proven in [12]. Another consequence of Theorem 1.1 is a result of Salwa, which states that if a and b satisfy the conditions of Theorem 1.1 and ab is non-nilpotent then (1 + a, (1 + b) m = 1 + mb) is a free pair for some positive integer m. In Section 5 we discuss the minimal m for which (1 + a, 1 + mb) is a free pair.…”

Section: Introductionmentioning

confidence: 93%

“…Their proof is not constructive and this raised the question of exhibiting a concrete free pair. This goal was achieved for non-Hamiltonian groups by Marciniak and Sehgal [12] using bicyclic units, and for Hamiltonian groups (non 2-group) by Ferraz [4] using Bass cyclic units. These results, together with a classical theorem of Bass [2], that states that if G is abelian then the Bass cyclic units generates a subgroups of finite index in U (ZG), and the more recent ones of Ritter and Sehgal [15] and Jespers and Leal [9], which prove that the group generated by the bicyclic and the Bass cyclic units generates a big portion of U (ZG), show that these two types of units have an important role in the structure of U (ZG).…”

Section: Introductionmentioning

confidence: 99%

Bicyclic units, Bass cyclic units and free groups

Gonçalves

Río

2008

Journal of Group Theory

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Let G be a finite group and ZG its integral group ring. We show that if α is a non-trivial bicyclic unit of ZG, then there are bicyclic units β and γ of different types, such that α, β and α, γ are non-abelian free groups. In case that G is non-abelian of order coprime with 6, then we prove the existence of a bicyclic unit u and a Bass cyclic unit v in ZG, such that for every positive integer m big enough, u m , v is a free non-abelian group. IntroductionA free pair is by definition a pair formed by two generators of a non-abelian free group. Let G be a finite group. The existence of free pairs in the group of units U (ZG) of the integral group ring ZG, was firstly proved by Hartley and Pickel [8], provided that G is neither abelian nor a Hamiltonian 2-group (equivalently U (ZG) is neither abelian nor finite , which prove that the group generated by the bicyclic and the Bass cyclic units generates a big portion of U (ZG), show that these two types of units have an important role in the structure of U (ZG). As a consequence, several authors have payed attention to the problem of describing the structure of the group generated either by bicyclic units, or Bass cyclic units and more specifically, to the problem of deciding when two bicyclic units or Bass cyclic units form a free pair [3,7,10,16]. The bicyclic units of ZG are the elements of one of the following forms

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“…An explicit construction of a free subgroup of the unit group was given by Marciniak and Sehgal in [12]: it is shown that any non-trivial bicyclic unit together with its image under the classical involution (which also is a bicylic unit) generate a non-cyclic free group. Since then many more constructions of two bicyclic units, or two Bass units, or a Bass together with a bicyclic unit generating a free group have been discovered.…”

Section: Introductionmentioning

confidence: 99%

Free products in the unit group of the integral group ring of a finite group

Janssens

Jespers

Temmerman

2017

Proc. Amer. Math. Soc.

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Abstract. Let G be a finite group and let p be a prime. We continue the search for generic constructions of free products and free monoids in the unit group U (ZG) of the integral group ring ZG. For a nilpotent group G with a non-central element g of order p, explicit generic constructions are given of two periodic units b 1 and b 2 in U (ZG) such that b 1 , b 2 = b 1 ⋆ b 2 ∼ = Zp ⋆ Zp, a free product of two cyclic groups of prime order. Moreover, if G is nilpotent of class 2 and g has order p n , then also concrete generators for free products Z p k ⋆ Z p m are constructed (with 1 ≤ k, m ≤ n). As an application, for finite nilpotent groups, we obtain earlier results of Marciniak-Sehgal and Gonçalves-Passman. Further, for an arbitrary finite group G we give generic constructions of free monoids in U (ZG) that generate an infinite solvable subgroup.

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Constructing free subgroups of integral group ring units

Abstract: Abstract. Let G be an arbitrary group. It is proved that if ZG contains a bicyclic unit u = 1, then u, u * is a nonabelian free subgroup of invertible elements.

Cited by 41 publications

References 5 publications

Free groups in quaternion algebras

Free groups in quaternion algebras

Bicyclic units, Bass cyclic units and free groups

Free products in the unit group of the integral group ring of a finite group

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