1997
DOI: 10.1090/s0002-9939-97-03812-4
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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.

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Cited by 41 publications
(9 citation statements)
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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%
“…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%
“…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%
“…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%