The description of the subgroup structure of a non-commutative division ring is the subject of the intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In this paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings.
Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian free subgroups in $N$. In particular, we show that if either $N$ is algebraic over $F$ or $F$ is uncountable, then $N$ contains a non-abelian free subgroup.
We give an example of a division ring $D$ whose multiplicative commutator subgroup does not generate $D$ as a vector space over its centre, thus disproving the conjecture posed in the paper "Vector space generated by the multiplicative commutators of a division ring, J. Algebra Appl. 12 (2013), no. 8".
We generalize Albert's twisted field construction, applying it to unital division algebras with a multiplicative norm. We give conditions for the resulting algebras to be division algebras.Four- and eight-dimensional real unital and non-unital division algebras with large derivation algebras are constructed out of Hamilton's quaternion and Cayley's octonion algebra.
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.
Let D be a division ring with centre F. Let T(D) be the vector space over F generated by all multiplicative commutators in D. In [1], authors have conjectured that every division ring is generated as a vector space over its centre by all of its multiplicative commutators. In this note it is shown that if D is centrally finite, then the conjecture holds.
Let $D$ be a division ring. In this paper, we investigate properties of subgroups of an arbitrary subnormal subgroup of the multiplicative group $D^*$ of $D$. The new obtained results generalize some previous results on subgroups of $D^*$.
The question of existence of a maximal subgroup in the multiplicative group D* of a division algebra D finite dimensional over its center F is investigated. We prove that if D* has no maximal subgroup, then deg(D) is not a power of 2, F^{*2} is divisible, and for each odd prime p dividing deg(D), there exist noncyclic division algebras of degree p over F.
Let $D$ be a division ring with the center $F$ and $D^*$ be the multiplicative group of $D$. In this paper we study locally nilpotent maximal subgroups of $D^*$. We give some conditions that influence the existence of locally nilpotent maximal subgroups in division ring with infinite center. Also, it is shown that if $M$ is a locally nilpotent maximal subgroup that is algebraic over $F$, then either it is the multiplicative group of some maximal subfield of $D$ or it is center by locally finite. If, in addition we assume that $F$ is finite and $M$ is nilpotent, then the second case cannot occu…
Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$. Among results, it is proved that if $D^*$ contains a noncentral normal subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le d^2$. In case $K=F$, the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of $D^*$ are algebraic of bounded degree $d$…
Let $D$ be a division ring with infinite center, $K$ a proper division subring of $D$ and $N$ an almost subnormal subgroup of the multiplicative group $D^*$ of $D$. The aim of this paper is to show that if $K$ is $N$-invariant and $N$ is non-central, then $K$ is central. Some examples of almost subnormal subgroups in division rings that are not subnormal are also given.
We introduce Kurosh elements in division rings based on the idea of a conjecture of Kurosh. Using this, we generalize a result of Faith in [3] and of Herstein in [6].
This paper, together with a forthcoming paper by the author and Seitz, proves the Margulis-Platonov conjecture concerning the normal subgroup structure of algebraic groups over number fields, in the case of inner forms of anisotropic groups of type $A_n$.
In this paper we prove that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite division ring contains free non-cyclic subgroups.
Let $D$ be a division ring with center $F$ and $K$ a division subring of $D$. In this paper, we show that a non-central normal subgroup $N$ of the multiplicative group $D^*$ is left algebraic over $K$ if and only if so is $D$ provided $F$ is uncountable and contained in $K$. Also, if $K$ is a field and the $n$-th derived subgroup $D^{(n)}$ of $D^{*}$ is left algebraic of bounded degree $d$ over $K$, then $\dim_FD\le d^2$.
Let $D$ be a division ring with center $F$, and $G$ a subnormal or quasinormal subgroup of $D^*$. We show that if $G$ is locally solvable, then $G$ is contained in $F$.
We investigate the finite subgroups that occur in the Hamiltonian quaternion algebra over the real subfield of cyclotomic fields. When possible, we investigate their distribution among the maximal orders.
Let $D$ be a division ring with center $F$. We say that $D$ is a {\em division ring of type $2$} if for every two elements $x, y\in D,$ the division subring $F(x, y)$ is a finite dimensional vector space over $F$. In this paper we investigate multiplicative subgroups in such a ring.
Let $D$ be a division ring with center $F$, and $G$ a subnormal subgroup of $D^*$. We show that if $G$ is a locally solvable group such that $G^{(i)}$ is algebraic over $F$, then $G$ must be central. Also, if $M$ is non-abelian locally solvable maximal subgroup of $G$ with $M^{(i)}$ algebraic over $F$, then $D$ is a cyclic algebra of prime degree over $F$.
In this paper we study non-central almost subnormal subgroups of the multiplicative group of a division ring satisfying a non-zero generalized rational identity. The main result generalizes Chiba's theorem on subnormal subgroups. As an application, we get a theorem on almost subnormal subgroups satisfying a generalized algebraic rational identity. The last theorem has several corollaries which generalize completely or partially some previous results.
Let $K$ be a field and let $σ$ be an automorphism and let $δ$ be a $σ$-derivation of $K$. Then we show that the multiplicative group of nonzero elements of the division ring $D=K(x;σ,δ)$ contains a free non-cyclic subgroup unless $D$ 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.
In this note, we introduce a new concept of a {\it generalized algebraic rational identity} to investigate the structure of division rings. The main theorem asserts that if a non-central subnormal subgroup $N$ of the multiplicative group $D^*$ of a division ring $D$ with center $F$ satisfies a non-trivial generalized algebraic rational identity of bounded degree, then $D$ is a finite dimensional vector space over $F$. This generalizes some previous results.
In Hai-Thin (2009), there is a theorem, stating that every locally nilpotent subnormal subgroup in a division ring $D$ is central (see Hai-Thin (2009, Th. 2.2)). Unfortunately, there is some mistake in the proof of this theorem. In this note we give the another proof of this theorem.
Meadows have been proposed as alternatives for fields with a purely equational axiomatization. At the basis of meadows lies the decision to make the multiplicative inverse operation total by imposing that the multiplicative inverse of zero is zero. Thus, the multiplicative inverse operation of a meadow is an involution. In this paper, we study `non-involutive meadows', i.e.\ variants of meadows in which the multiplicative inverse of zero is not zero, and pay special attention to non-involutive meadows in which the multiplicative inverse of zero is one.
Let $D$ be a division ring, $n$ a positive integer, and GL$_n(D)$ the general linear group of degree $n$ over $D$. In this paper, we study the induced subgraph of the intersection graph of GL$_n(D)$ generated by all non-trivial proper almost subnormal subgroups of GL$_n(D)$. We show that this subgraph is complete if it is non-null. This property will be used to study subgroup structure of a division ring. In particular, we prove that every non-central almost subnormal subgroup of the multiplicative group $D^*$ of a division ring $D$ contains a non-central subnormal subgroup of $D^*$.
General results on multiplicative lattices found recently by Facchini, Finocchiaro and Janelidze have been studied in the particular case of groups by Facchini, de Giovanni and Trombetti. In this paper we prove that these results hold not only for the multiplicative lattices of all normal subgroups of a group, but also for much more general multiplicative lattices. Therefore they can be applied to other algebraic structures, for instance to braces.
Let $D$ be a division ring infinite-dimensional over its center $k$ with multiplicative group $D^{\times}$. We show that if $D$ belongs to certain families, there exist free symmetric and unitary pairs in $D^{\times}$ with respect to a $k$-involution on $D$.
In this paper, we prove that the multiplicative group of a unital non-commutative Leavitt path algebra $L_K(E)$ and Cohn path algebra $C_K(E)$ contain a non-cyclic free subgroup, provided $K$ is a non-absolute field. We also provide a description of the generators of free subgroups in term of the graph $E$. Finally, we determine multiplicative groups of Leavitt path algebras of some special types.
Graded-division algebras are building blocks in the theory of finite-dimensional associative algebras graded by a group G. If G is abelian, they can be described, using a loop construction, in terms of central simple graded-division algebras. On the other hand, given a finite abelian group G, any central simple G-graded-division algebra over a field F is determined, thanks to a result of Picco and Platzeck, by its class in the (ordinary) Brauer group of F and the isomorphism class of a G-Galois extension of F. This connection is used to classify the simple G-Galois extensions of F in terms of …
In this paper, we give the definition of {\em weakly locally finite} division rings and we show that the class of these rings strictly contains the class of locally finite division rings. Further, we study multiplicative subgroups in these rings. Some skew linear groups are also considered. Our new obtained results generalize previous results for centrally finite case.
Finite groups that are embeddable in the multiplicative groups of division rings $K$ were completely determined by S. A. Amitsur in 1955. In case $K$ has characteristic $p>0$, the only possible finite subgroups of $K^*$ are cyclic groups, according to a theorem of I. N. Herstein. Thus, the only interesting case is when $K$ has characteristic 0; that is, when $K\supseteq {\Bbb Q}$. Herstein conjectured that odd-order subgroups of division rings $K$ were cyclic, and he proved this to be the case when $K$ is the division ring of the real quaternions. Herstein's conjecture was settled negatively b…
Let $D$ be a non-commutative division ring, $G$ a subnormal subgroup of ${\mathrm GL}_n(D)$. In this note we show that if $G$ contains a non-abelian solvable maximal subgroup, then $n=1$ and $D$ is a cyclic algebra of prime degree over $F$.
Let D be a division ring finite dimensional over its center F. The goal of this paper is to prove that for any positive integer n there exists a in D^(n); the n-th multiplicative derived subgroup, such that F(a) is a maximal subfield of D. We also show that a single depth-n iterated additive commutator would generate a maximal subfield of D.
We give an explicit list of all p-groups G with a cyclic subgroup of index p^2, such that the group algebra KG over the field K of characteristic p has a filtered multiplicative K-basis. We also proved that such a K-basis does not exist for the group algebra KG, in the case when G$ is either a powerful p-group or a two generated p-group (p\not=2) with a central cyclic commutator subgroup. This paper is a continuation of the related V. Bovdi, On a filtered multiplicative basis of the group algebras Arch. Math. (Basel) 74 (2000) 81--88
In this paper, we study almost subnormal subgroups of the general linear group $\GL_n(D)$ of degree $n\ge 1$ over a division ring $D$ that satisfy a generalized power central group identity.
Let $D$ be a weakly locally finite division ring and $n$ 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 ${\rm GL}_n(D)$. Further, some applications of this fact are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of ${\rm GL}_n(D)$ are described.
Any superrosy division ring (i.e. a division ring equipped with an abstract notion of rank) is shown to be centrally finite. Furthermore, division rings satisfying a generalized chain condition on definable subgroups are studied. In particular, a division ring of burden n has dimension at most n over its center, and any definable group of definable automorphisms of a field of burden n has size at most n. Additionally, interpretable division rings in o-minimal structures are shown to be algebraically closed, real closed or the quaternions over a real closed field.
We determine the algebraic structure of the multiplicative loops for locally compact $2$-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a $1$-dimensional compact subgroup. In the last section we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension $4$ admitting an at least $7$-dimensional Lie group as collineation group.
Let $R$ be a finite-dimensional algebra over an algebraically closed field $F$ graded by an arbitrary group $G$. We prove that $R$ is a graded division algebra if and only if it is isomorphic to a twisted group algebra of some finite subgroup of $G$. If the characteristic of $F$ is zero or ${\rm char} F$ does not divide the order of any finite subgroup of $G$ then we prove that $R$ is graded simple if and only if it is a matrix algebra over a finite-dimensional graded division algebra.
This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group $\GL_n(D)$ over a division ring $D$. It turns out that in the case where $D$ is non-commutative, if such maximal subgroups exist, then either it is abelian or $[D:F]<\infty$. Also, if $F$ is an infinite field and $n\geq 5$, then every locally solvable maximal subgroup of a normal subgroup of $\GL_n(F)$ is abelian.
Let $D$ be a division ring with infinite center, $n$ be a positive integer and $w(x_1,x_2,\cdots, x_m)=1$ be a generalized group identity over the general linear group $\GL_n(D)$. The aim of this paper is to prove that every subnormal subgroup of $GL_n(D)$ which satisfies the generalized group identity $w(x_1,x_2,\cdots, x_m)=1$ is central.
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.
Let $D$ be a division ring with the center $F=Z(D)$. Suppose that $N$ is a normal subgroup of $D^*$ which is radical over $F$, that is, for any element $x\in N$, there exists a positive integer $n_x$, such that $x^{n_x}\in F$. In \cite{Her1}, Herstein conjectured that $N$ is contained in $F$. In this paper, we show that the conjecture is true if there exists a positive integer $d$ such that $n_x\le d$ for any $x\in N$
Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a normal abelian subgroup $H$, a positive integer $d$ and an element of $Hom(M(H), k^\times)^G$, where $M(H)$ is the Schur multiplier of $H$. Our main theorem is the converse: Given an extension $1\rightarrow H\rightarrow G\rightarrow G/H\rightarrow 1$, where $H$ is abelian, a positive integer $d$, and an element of $Hom(M(H), k^\times)^G$, there is a division al…
Let $\mathbb{k}$ be an algebraically closed field of characteristic zero. Let $D$ be a division algebra of degree $d$ over its center $Z(D)$. Assume that $\mathbb{k}\subset Z(D)$. We show that a finite group $G$ faithfully grades $D$ if and only if $G$ contains a normal abelian subgroup of index dividing $d$. We also prove that if a finite dimensional Hopf algebra coacts on $D$ defining a Hopf-Galois extension, then its PI degree is at most $d^2$. Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles.
