4,261 research outputs found
Free subgroups in almost subnormal subgroups of general skew linear groups
Let be a weakly locally finite division ring and 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
. Further, some applications of this fact are also investigated.
In particular, all infinite finitely generated almost subnormal subgroups of
are described.Comment: 12 page
Multiplicative groups of division rings
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
Let be a division ring. In this paper, we investigate properties of
subgroups of an arbitrary subnormal subgroup of the multiplicative group
of . The new obtained results generalize some previous results on subgroups
of .Comment: 14 page
Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups
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
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
Let be a field of characteristic different from and let be a
nonabelian residually torsion-free nilpotent group. It is known that is an
orderable group. Let denote the subdivision ring of the Malcev-Neumann
series ring generated by the group algebra of over . If is an
involution on , then it extends to a unique -involution on . We
show that contains pairs of symmetric elements with respect to
which generate a free group inside the multiplicative group of . Free
unitary pairs also exist if is torsion-free nilpotent. Finally, we consider
the general case of a division ring , with a -involution ,
containing a normal subgroup in its multiplicative group, such that , with a nilpotent-by-finite torsion-free subgroup that is not
abelian-by-finite, satisfying and . We prove that
contains a free symmetric pair.Comment: 14 page
On locally solvable subgroups in division rings
Let be a division ring with center , and a subnormal subgroup of
. We show that if is a locally solvable group such that is
algebraic over , then must be central. Also, if is non-abelian
locally solvable maximal subgroup of with algebraic over ,
then is a cyclic algebra of prime degree over
On free subgroups in division rings
Let be a field and let be an automorphism and let be a
-derivation of . Then we show that the multiplicative group of
nonzero elements of the division ring contains a free
non-cyclic subgroup unless 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
Let be a division ring with center and a subnormal subgroup of
the multiplicative group of . Assume that contains a non-abelian
solvable subgroup. In this paper, we study the problem on the existence of
non-abelian free subgroups in . In particular, we show that if either is
algebraic over or is uncountable, then contains a non-abelian free
subgroup.Comment: 10 page
Free algebras and free groups in Ore extensions and free group algebras in division rings
Let be a field of characteristic zero, let be an automorphism of
and let be a -derivation of . We show that the division
ring either has the property that every finitely
generated subring satisfies a polynomial identity or contains a free
algebra on two generators over its center. In the case when is finitely
generated over we then see that for a -algebra automorphism of
and a -linear derivation of , having a free
subalgebra on two generators is equivalent to having infinite order,
and having a free subalgebra is equivalent to being
nonzero. As an application, we show that if is a division ring with center
of characteristic zero and contains a solvable subgroup that is not
locally abelian-by-finite, then contains a free -algebra on two
generators. Moreover, if we assume that is uncountable, without any
restrictions on the characteristic of , then contains the -group
algebra of the free group of rank two.Comment: 13 page
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