4,261 research outputs found

    Free subgroups in almost subnormal subgroups of general skew linear groups

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    Let DD be a weakly locally finite division ring and nn a positive integer. In this paper, we investigate the problem on the existence of non-cyclic free subgroups in non-central almost subnormal subgroups of the general linear group GLn(D){\rm GL}_n(D). Further, some applications of this fact are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of GLn(D){\rm GL}_n(D) are described.Comment: 12 page

    Multiplicative groups of division rings

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    Exactly 170 years ago, the construction of the real quaternion algebra by William Hamilton was announced in the Proceedings of the Royal Irish Academy. It became the first example of non-commutative division rings and a major turning point of algebra. To this day, the multiplicative group structure of quaternion algebras have not completely been understood. This article is a long survey of the recent developments on the multiplicative group structure of division rings

    On multiplicative subgroups in division rings

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    Let DD be a division ring. In this paper, we investigate properties of subgroups of an arbitrary subnormal subgroup of the multiplicative group D∗D^* of DD. The new obtained results generalize some previous results on subgroups of D∗D^*.Comment: 14 page

    Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups

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    Let D be an infinite division ring, n a natural number and N a subnormal subgroup of GLn(D) such that n = 1 or the center of D contains at least five elements. This paper contains two main results. In the first one we prove that each nilpotent maximal subgroup of N is abelian; this generalizes the result in [R. Ebrahimian, J. Algebra 280 (2004) 244 - 248] (which asserts that each max- imal subgroup of GLn(D) is abelian) and a result in [M. Ramezan-Nassab, D. Kiani, J. Algebra 376 (2013) 1 - 9]. In the second one we show that a maximal subgroup of GLn(D) cannot be polycyclic-by-finite.Comment: 9 page

    Ring-theoretic properties of Iwasawa algebras: a survey

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    This is a survey of the known properties of Iwasawa algebras, which are completed group rings of compact p-adic analytic groups with coefficients the ring Zp of p-adic integers or the field Fp of p elements. A number of open questions are also stated.Comment: 27 pages, submitted to the John Coates volume of Documenta Mathematic

    Free symmetric and unitary pairs in the field of fractions of torsion-free nilpotent group algebras

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    Let kk be a field of characteristic different from 22 and let GG be a nonabelian residually torsion-free nilpotent group. It is known that GG is an orderable group. Let k(G)k(G) denote the subdivision ring of the Malcev-Neumann series ring generated by the group algebra of GG over kk. If ∗\ast is an involution on GG, then it extends to a unique kk-involution on k(G)k(G). We show that k(G)k(G) contains pairs of symmetric elements with respect to ∗\ast which generate a free group inside the multiplicative group of k(G)k(G). Free unitary pairs also exist if GG is torsion-free nilpotent. Finally, we consider the general case of a division ring DD, with a kk-involution ∗\ast, containing a normal subgroup NN in its multiplicative group, such that G⊆NG \subseteq N, with GG a nilpotent-by-finite torsion-free subgroup that is not abelian-by-finite, satisfying G∗=GG^{*}=G and N∗=NN^{*}=N. We prove that NN contains a free symmetric pair.Comment: 14 page

    On locally solvable subgroups in division rings

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    Let DD be a division ring with center FF, and GG a subnormal subgroup of D∗D^*. We show that if GG is a locally solvable group such that G(i)G^{(i)} is algebraic over FF, then GG must be central. Also, if MM is non-abelian locally solvable maximal subgroup of GG with M(i)M^{(i)} algebraic over FF, then DD is a cyclic algebra of prime degree over FF

    On free subgroups in division rings

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    Let KK be a field and let σ\sigma be an automorphism and let δ\delta be a σ\sigma-derivation of KK. Then we show that the multiplicative group of nonzero elements of the division ring D=K(x;σ,δ)D=K(x;\sigma,\delta) contains a free non-cyclic subgroup unless DD is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free non-abelian solvable-by-finite groups always contain free non-cyclic subgroups.Comment: nine page

    On subnormal subgroups in division rings containing a non-abelian solvable subgroup

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    Let DD be a division ring with center FF and NN a subnormal subgroup of the multiplicative group D∗D^* of DD. Assume that NN contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in NN. In particular, we show that if either NN is algebraic over FF or FF is uncountable, then NN contains a non-abelian free subgroup.Comment: 10 page

    Free algebras and free groups in Ore extensions and free group algebras in division rings

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    Let KK be a field of characteristic zero, let σ\sigma be an automorphism of KK and let δ\delta be a σ\sigma-derivation of KK. We show that the division ring D=K(x;σ,δ)D=K(x;\sigma,\delta) either has the property that every finitely generated subring satisfies a polynomial identity or DD contains a free algebra on two generators over its center. In the case when KK is finitely generated over kk we then see that for σ\sigma a kk-algebra automorphism of KK and δ\delta a kk-linear derivation of KK, K(x;σ)K(x;\sigma) having a free subalgebra on two generators is equivalent to σ\sigma having infinite order, and K(x;δ)K(x;\delta) having a free subalgebra is equivalent to δ\delta being nonzero. As an application, we show that if DD is a division ring with center kk of characteristic zero and D∗D^* contains a solvable subgroup that is not locally abelian-by-finite, then DD contains a free kk-algebra on two generators. Moreover, if we assume that kk is uncountable, without any restrictions on the characteristic of kk, then DD contains the kk-group algebra of the free group of rank two.Comment: 13 page
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