Buildings and classical groups

Projective bases of division algebras and groups of central type II

Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra A if it is isomorphic to a twisted group algebra k^αG for some class αin H^2(G,k^\times), where the action of G on k^\times is trivial. In a preceding paper by Aljadeff, Haile and the author (Projective bases of division algebras and groups of central type, Israel J. Math. 146 (2005) 317-335) it was shown that if a group G is a projective basis in a k-central division algebra then G is nilpotent and every Sylow-p subgroup of G is on the short list of families of p-groups, denoted by Λ. In this pape…

Moufang sets and structurable division algebras

A Moufang set is essentially a doubly transitive permutation group such that each point stabilizer contains a normal subgroup which is regular on the remaining vertices; these regular normal subgroups are called the root groups, and they are assumed to be conjugate and to generate the whole group. It has been known for some time that every Jordan division algebra gives rise to a Moufang set with abelian root groups. We extend this result by showing that every structurable division algebra gives rise to a Moufang set, and conversely, we show that every Moufang set arising from a simple linear a…

Idempotent solutions of the Yang-Baxter equation and twisted group division

Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using combinatorial properties of the Cayley kernel and the squaring mapping, we prove that a twisted Ward left quasigroup of prime order is either permutational or a quasigroup. Up to isomorphism, all twisted Ward quasigroups $(X,*)$ are obtained by twisting the left division operation in groups (that is, they are of the form $x*y=ψ(x^{-1}y)$ for a group $(X,\cdot)$ and its automor…

Moufang sets arising from polarities of Moufang planes over octonion division algebras

For every octonion division algebra O, there exists a projective plane which is parametrized by O; these planes are related to rank two forms of linear algebraic groups of absolute type E6. We study all possible polarities of such octonion planes having absolute points, and their corresponding Moufang set. It turns out that there are four different types of polarities, giving rise to (1) Moufang sets of type F4, (2) Moufang sets of type 2E6, (3) hermitian Moufang sets of type C4, and (4) projective Moufang sets over a 5-dimensional subspace of an octonion division algebra. Case (3) only occurs…

赋值除代数的阿贝尔极大子群

Abelian maximal subgroups of valued division algebras

研究除环乘法群何时包含阿贝尔极大子群，证明在非平凡Hensel赋值域上的非交换有限维中心除代数中不可能存在，并推广到标准赋值除环，最后利用补malnormality障碍排除实闭域上四元数除代数的阿贝尔极大子群。

Hyperbolic Weyl groups and the four normed division algebras

We study the Weyl groups of hyperbolic Kac-Moody algebras of `over-extended' type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed division algebras K=R,C,H,O, respectively. A crucial role is played by integral lattices of the division algebras and associated discrete matrix groups. Our findings can be summarized by saying that the even subgroups, W^+, of the Kac-Moody Weyl groups, W, are isomorphic to generalized modular groups over K for the simply laced algebras, and to certain finite extensions thereof for the non-simply laced algebras. This hints at an extended t…

Small generators for S-unit groups of division algebras

Let $k$ be a number field, suppose that $B$ is a central simple division algebra over $k$, and choose any maximal order $\mathcal{D}$ of $B$. The object of this paper is to show that the group $\mathcal{D}_S^*$ of $S$-units of $B$ is generated by elements of small height once $S$ contains an explicit finite set of places of $k$. This generalizes a theorem of H.\ W.\ Lenstra Jr., who proved such a result when $B = k$. Our height bound is an explicit function of the number field and the discriminant of a maximal order in $B$ used to define its $S$-units.

On the second cohomology of the norm one group of a p-adic division algebra

Let $F$ be a $p$-adic field, that is, a finite extension of $\mathbb Q_p$. Let $D$ be a finite-dimensional central division algebra over $F$ and let $SL_1(D)$ be the group of elements of reduced norm $1$ in $D$. Prasad and Raghunathan proved that $H^2(SL_1(D),\mathbb R/\mathbb Z)$ is a cyclic $p$-group whose order is bounded from below by the number of $p$-power roots of unity in $F$, unless $D$ is a quaternion algebra over $\mathbb Q_2$. In this paper we give an explicit upper bound for the order of $H^2(SL_1(D),\mathbb R/\mathbb Z)$ for $p\geq 5$ and determine $H^2(SL_1(D),\mathbb R/\mathbb …

Multiplicative Subgroups in weakly locally finite division rings

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.

On the self-similarity of the norm one group of $p$-adic division algebras

Let $p$ be a prime, $D$ a finite dimensional noncommutative division $\mathbb{Q}_p$-algebra, and $SL_1(D)$ the group of elements of $D$ of reduced norm $1$. When the center of $D$ is $\mathbb{Q}_p$, we prove that no open subgroup of $SL_1(D)$ admits self-similar actions on regular rooted trees. Moreover, we prove results on $\mathbb{Z}_p$-Lie lattices that allow to deal with the case where the center of $D$ is bigger than $\mathbb{Q}_p$, and lead to the classification of the torsion-free $p$-adic analytic pro-$p$ groups $G$ of dimension less than $p$ with the property that all the nontrivial c…

Some results relevant to embeddability of rings (especially group algebras) in division rings

P. M. Cohn showed in 1971 that given a ring $R$, to describe, up to isomorphism, a division ring $D$ generated by a homomorphic image of $R$ is equivalent to specifying the set of square matrices over $R$ which map to singular matrices over $D,$ and he determined precisely the conditions that such a set of matrices must satisfy. The present author later developed another version of this data, in terms of closure operators on free $R$-modules. In this note, we examine the latter concept further, and show how an $R$-module $M$ satisfying certain conditions can be made to induce such data. In an …

Graph Invariants of Finite Groups via a Theorem of Lagarias

We introduce a new graph invariant of finite groups that provides a complete characterization of the splitting types of unramified prime ideals in normal number field extensions entirely in terms of the Galois group. In particular, each connected component corresponds to a division (Abteilung) of the group. We compute the divisions of the alternating group, and compile a list of characteristics of groups that the invariant reveals. We conjecture that the invariant distinguishes finite groups. Our exposition borrows elements from graph theory, group theory, and algebraic number theory.

定义在数域上的椭圆曲线的除法域重合

Coincidences of Division Fields of an elliptic curve defined over a number field

本文研究数域上椭圆曲线由不同阶挠点生成的除法域何时相等，分别给出纵向与横向重合的必要条件，并构造非 Q 上的非平凡例子。

分次赋值除环的群$\mathrm{TK}_1$

The group $\mathrm{TK}_1$ of graded and valued division algebras

研究分次赋值除环的挠群$\mathrm{TK}_1$，给出描述$\mathrm{TK}_1(E)$的正合序列及显式公式，确定阻碍群$\mathbf{H}$，证明短正合序列和稳定性定理，还得到挠怀特黑德群相关结果的分次类似物。

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

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.

Conjugacy in Baumslag's group, generic case complexity, and division in power circuits

The conjugacy problem belongs to algorithmic group theory. It is the following question: given two words x, y over generators of a fixed group G, decide whether x and y are conjugated, i.e., whether there exists some z such that zxz^{-1} = y in G. The conjugacy problem is more difficult than the word problem, in general. We investigate the complexity of the conjugacy problem for two prominent groups: the Baumslag-Solitar group BS(1,2) and the Baumslag(-Gersten) group G(1,2). The conjugacy problem in BS(1,2) is TC^0-complete. To the best of our knowledge BS(1,2) is the first natural infinite no…

关于赋值可除代数的同余定理

On Congruence Theorem for valued division algebras

本文研究 Hensel 赋值域上驯顺中心可除代数的 $\mathrm{TK}_1(D)$，证明其与主单位群相关部分要么平凡、要么在正特征剩余域下为 $p$-群，并用反例说明同余定理不能完全推广。

On group rings of virtually abelian groups

Let $Γ$ be a finitely generated torsion-free group. We show that the statement of $Γ$ being virtually abelian is equivalent to the statement that the $*$-regular closure of the group ring $\mathbb{C}[Γ]$ in the algebra of (unbounded) operators affiliated to the group von Neumann algebra is a central division algebra. More generally, for any field $k$, it is shown that $k[Γ]$ embeds into a central division algebra in case $Γ$ is virtually abelian. We take advantage of this result in order to develop a criterion for existence of units in the group ring $k[Γ]$. We develop this criterion in the pa…

A classification of exceptional components in group algebras over abelian number fields

When considering the unit group of $\mathcal{O}_F G$ ($\mathcal{O}_F$ the ring of integers of an abelian number field $F$ and a finite group $G$) certain components in the Wedderburn decomposition of $FG$ cause problems for known generic constructions of units; these components are called exceptional. Exceptional components are divided into two types: type 1 are division rings, type 2 are $2 \times 2$-matrix rings. For exceptional components of type 1 we provide infinite classes of division rings by describing the seven cases of minimal groups (w.r.t. quotients) having those division rings in …