Yoav Segev

Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let D be a finite dimensional division algebra having center K and let N ⊆ D× be a normal subgroup of finite index. Suppose D×/N is not solvable. Then we may assume that H := D×/N is a minimal nonsolvable group (MNS group for short), i.e., a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote property (3 12 ). This property includes the requirement that the diameter of the commuting graph should be ≥ 3, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of D×/N has the property (312 ), then N is open with respect to a nontrivial height one valuation of D (assuming without loss, as we may, that K is finitely generated). After establishing the openness of N (when D×/N is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of K over its prime subfield, to eliminate H as a possible quotient of D×, thereby obtaining a contradiction and proving our main result.

The Commuting Graph of Minimal Nonsolvable Groups

The purpose of this paper is to prove that if G is a finite minimal nonsolvable group (i.e. G is not solvable but every proper quotient of G is solvable), then the commuting graph of G has diameter ≥3. We give an example showing that this result is the best possible. This result is related to the structure of finite quotients of the multiplicative group of a finite-dimensional division algebra.

Identities in Moufang sets

Moufang sets were introduced by Jacques Tits as an axiomatization of the buildings of rank one that arise from simple algebraic groups of relative rank one. These fascinating objects have a simple definition and yet their structure is rich, while it is rigid enough to allow for (at least partial) classification. In this paper we obtain two identities that hold in any Moufang set. These identities are closely related to the axioms that define a quadratic Jordan algebra. We apply them in the case when the Moufang set is so-called special and has abelian root groups. In addition we push forward the theory of special Moufang sets.

On the action of the Hua subgroups in special Moufang sets

We show that in a special Moufang set, either the root groups are elementary abelian 2-groups, or the Hua subgroup H (= the Cartan subgroup) acts ?irreducibly? on U, i.e. the only non-trivial H-invariant subgroup of a root group normalized by H is the whole root group.

Proper Moufang sets with abelian root groups are special

Moufang sets are split BN -pairs of rank one, or the Moufang buildings of rank one. As such they have been studied extensively being the basic ‘building blocks’ of all split BN -pairs. A Moufang set is proper if it is not sharply 2-transitive. We prove that a proper Moufang set whose root groups are abelian is special. This resolves an important conjecture in the area of Moufang sets. It enables us to apply the theory of quadratic Jordan division algebras to such Moufang sets.

Finite special Moufang sets of even characteristic

In this paper we classify finite special Moufang sets 𝕄(U,τ) of odd characteristic. The characteristic 2 case was handled in another paper by De Medts and the author (see [3]). We prove that U is elementary abelian. Then, we show that 𝕄(U,τ) is the unique Moufang set whose little projective group is PSL2(|U|). The emphasis of this paper is on obtaining elementary proofs. Section 3 deals with root subgroups in any Moufang set and may be of independent interest.

The uniqueness of groups of Lyons type

We give the first computer free proof of the uniqueness of groups of type Ly. We also show that certain simpicial complexes associated to the Lyons group Ly have the same homotopy type as the Quillen complex for Ly at the prime 3; we show this complex is simply connected, and we calculate its homology. Finally we supply simplified proofs of some properties of groups of type Ly, such as the group order. A group of type Ly is a finite group G possessing an involution t such that H = CG(t) is the covering group of the alternating group A1l and t is not weakly closed in H with respect to G. We prove

Locally connected simplicial maps

In Propositions 1.6 and 7.6 of his paper onp-group complexes of finite groups [5], Quillen establishes fundamental results comparing the homology and the fundamental group of the order complexes of posetsP, Q admitting a mapf :P →Q of posets with good local behavior. We prove the analogue of Quillen’s results for mapsf :K→L of simplicial complexesK andL in a more general setup.

Group actions on finite acyclic simplicial complexes

In this paper we develop some homological techniques to obtain fixed points for groups acting on finite Z-acyclic complexes. In particular we show that if a groupG acts on a finite 2-dimensional acyclic simplicial complexD, then the fixed point set ofG onD is either empty or acyclic. We supply some machinery for determining which of the two cases occurs. The Feit-Thompson Odd Order Theorem is used in obtaining this result.

The finite quotients of the multiplicative group of a division algebra of degree 3 are solvable

LetD be a finite dimensional division algebra. It is known that in a variety of cases, questions about the normal subgroup structure ofDx (the multiplicative group ofD) can be reduced to questions about finite quotients ofDx. In this paper we prove that when deg(D)=3, finite quotients ofDx are solvable. the proof uses Wedderburn’s Factorization Theorem.