Let $F$ be a perfect field and $M^*(F)$ the nonassociative simple Moufang loop consisting of the units in the (unique) split octonion algebra $O(F)$ modulo the center. Then $Aut(M^*(F))$ is equal to $G_2(F) \rtimes Aut(F)$. In particular, every automorphism of $M^*(F)$ is induced by a semilinear automorphism of $O(F)$. The proof combines results and methods from geometrical loop theory, groups of Lie type and composition algebras; its gist being an identification of the automorphism group of a Moufang loop with a subgroup of the automorphism group of the associated group with triality.
We classify the automorphisms of a Moufang hexagon mapping no chamber to an opposite chamber (such automorphisms are called domestic). This forms part of a larger program to classify domestic automorphisms of Moufang spherical buildings.
Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F_4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Moufang quadrangle of type F_4 stabilizing the absolute points of a polarity, also centralizes the polarity. This handles the final case of a similar result for all polarities of Moufang n-gons.
We prove a purely topological characterization of the Moufang property for disconnected compact polygons in terms of convergence groups. As a consequence, we recover the fact that a locally finite thick affine building of rank 3 is a Bruhat-Tits building if and only if its automorphism group is strongly transitive. We also study automorphism groups of general compact polygons without any homogeneity assumption. A compactness criterion for sets of automorphisms is established, generalizing the theorem by Burns and Spatzier that the full automorphism group, endowed with the compact-open topology…
It is well known that if a group G factorizes as G = NH where H\leq G and N is normal in G then the group structure of G is determined by the subgroups H and N, the intersection of N with H and how H acts on N with a homomorphism f : H -> Aut(N). Here we generalize the idea by creating extensions using the semi-automorphism group of N. We show that if G=NH is a Moufang loop, N is a normal subloop, and H=<u> is a finite cyclic group of order coprime to three then the binary operation of G depends only on the binary operation of N, the intersection of N with H, and how u permutes the elements of…
We show that each half-automorphism of a finite automorphic Moufang loop is trivial. In general this is not true for finite left automorphic Moufang loops and for finite automorphic loops.
An A-loop is a loop in which every inner mapping is an automorphism. We settle a problem which had been open since 1956 by showing that every diassociative A-loop is Moufang.
We prove that a normal subloop $X$ of a Moufang loop $Q$ induces an abelian congruence of $Q$ if and only if each inner mapping of $Q$ restricts to an automorphism of $X$ and $u(xy) = (uy)x$ for all $x,y\in X$ and $u\in Q$. The former condition can be omitted when $X$ is $3$-divisible. This characterization is then used to show that classically solvable finite $3$-divisible Moufang loops are congruence solvable.
Nonassociative finite simple Moufang loops are exactly the loops constructed by Paige from Zorn vector matrix algebras. We prove this result anew, using geometric loop theory. In order to make the paper accessible to a broader audience, we carefully discuss the connections between composition algebras, simple Moufang loops, simple Moufang 3-nets, $S$-simple groups and groups with triality. Related results on multiplication groups, automorphisms groups and generators of Paige loops are provided.
Sh. Mozes showed that the commensurator of the lattice ${\rm PSL}_2 \bigl({\bf F}_p[t{}^{-1}] \bigr)$ is dense in the full automorphism group of the Bruhat-Tits tree of valency $p+1$, the latter group being much bigger than ${\rm PSL}_2 \bigl({\bf F}_p((t)) \bigr)$. By G.A. Margulis' criterion, this density is a generalized arithmeticity result. We show that the density of the commensurator holds for many tree-lattices among those called of Nagao type by H. Bass and A. Lubotzky. The result covers many lattices obtained via Moufang twin trees.
In this work we construct free Moufang loop in the variety generated by code loops. We apply this construction for study the code loops. Moreover, we define and determine all basic representations of code loops of rank 3 and 4.
We study closed subgroups $G$ of the automorphism group of a locally finite tree $T$ acting doubly transitively on the boundary. We show that if the stabiliser of some end is metabelian, then there is a local field $k$ such that $\mathrm{PSL}_2(k) \leq G \leq \mathrm{PGL}_2(k)$. We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if $G$ is (virtually) a rank one simple $p$-adic analytic group for some prime $p$. A key point is that if some contraction group is closed, then $G$ is boundary-Moufang, meaning that the boundary $\partial T$ is a…
We study abelian-by-cyclic Moufang loops. We construct all split $3$-divisible abelian-by-cyclic Moufang loops from so-called Moufang permutations on abelian groups $(X,+)$, which are permutations that deviate from an automorphism of $(X,+)$ by an alternating biadditive mapping (satisfying certain properties). More generally, we obtain additional abelian-by-cyclic Moufang loops from so-called construction pairs. As an aside, we show that in the Moufang loops $Q$ obtained from a construction pair on $(X,+)$ the abelian normal subgroup $(X,+)$ induces an abelian congruence of $Q$ if and only if …
The fixed point building of a polarity of a Moufang quadrangle of type $F_4$ is a Moufang set, as is the fixed point building of a semi-linear automorphism of order $2$ of a Moufang octagon that stabilizes at least two panels of one type but none of the other. We show that these two classes of Moufang sets are, in fact, the same, that each member of this class can be constructed as the fixed point building of a group of order $4$ acting on a building of type $F_4$ and that the group generated by all the root groups of any one of these Moufang sets is simple.
T. De Medts, Y. Segev and K. Tent [Special Moufang sets, their root groups and their μ-maps, Proc. Lond. Math. Soc. (3) 96 (2008), 767-791] proved that the little projective group of a special Moufang set M(U,τ) is perfect provided that U has size at least 4. We give a short, elementary and self-contained argument, working with Timmesfeld's concept of special (abstract) rank one groups with arbitrary unipotent subgroups [Abstract root subgroups and simple groups of Lie type, Monographs in Mathematics, 95. Birkhäuser, 2001].
A bijection $f$ of a loop $L$ is a half-automorphism if $f(xy)\in \{f(x)f(y),f(y)f(x)\}$, for any $x,y\in L$. A half-automorphism is nontrivial when it is neither an automorphism nor an anti-automorphism. A Chein loop $L=G\cup Gu$ is a Moufang loop constructed from a group $G$ and an element $u$ of order $2$ outside of $G$. In this paper, the half-automorphism group of finite Chein loops is described.
T. De Medts, Y. Segev and K. Tent [Special Moufang sets, their root groups and their μ-maps, Proc. Lond. Math. Soc. (3) 96 (2008), 767-791] proved that the little projective group of a special Moufang set M(U,τ) is perfect provided that U has size at least 4. We give a short, elementary and self-contained argument when U is abelian, working with Timmesfeld's concept of special (abstract) rank one groups with abelian unipotent subgroups [Abstract root subgroups and simple groups of Lie type, Monographs in Mathematics, 95. Birkhäuser, 2001]. This simplifies Timmesfeld's (quasi) simplicity criter…
Let $M^*(q)$ be the unique nonassociative finite simple Moufang loop constructed over $GF(q)$. We prove that $Aut(M^*(2))$ is the Chevalley group $G_2(2)$, by extending multiplicative automorphism of $M^*(2)$ into linear automorphisms of the unique split octonion algebra over GF(2). Many of our auxiliary results apply in the general case. In the course of the proof we show that every element of a split octonion algebra can be written as a sum of two elements of norm one.
Firstly, we prove that every closed subgroup $H$ of type-preserving automorphisms of a locally finite thick affine building $Δ$ of dimension $\geq 2$ that acts strongly transitively on $Δ$ is Moufang. If moreover $Δ$ is irreducible and $H$ is topologically simple, we show that $H$ is the subgroup $\G(k)^+$ of the $k$-rational points $\G(k)$ of the isotropic simple algebraic group $\G$ over a non-Archimedean local field $k$ associated with $Δ$. Secondly, we generalise the proof given in \cite{BM00b} for the case of bi-regular trees to any locally finite thick affine building $Δ$, and obtain tha…
Let $k$ be an arbitrary field. The main aim of this paper is to prove the Tits-Weiss conjecture for Albert division algebras over $k$ which are pure first Tits constructions. This conjecture asserts that for an Albert division algebra $A$ over a field $k$, every norm similarity of $A$ is inner modulo scalar multiplications. It is known that $k$-forms of $E_8$ with index $E^{78}_{8,2}$ and anisotropic kernel a strict inner $k$-form of $E_6$ correspond bijectively (via Moufang hexagons) to Albert division algebras over $k$. The Kneser-Tits problem for a form of $E_8$ as above is equivalent to th…
Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes, for instance, groups and commutative Moufang loops. We study uniquely 2-divisible automorphic loops, particularly automorphic loops of odd order, from the point of view of the associated Bruck loops (motivated by Glauberman's work on uniquely 2-divisible Moufang loops) and the associated Lie rings (motivated by a construction of Wright). We prove that every automorphic loop $Q$ of odd order is solvable, contains an element of order $p$ for every prime $p$ dividing $|Q|$, and $|S|$ divides…
Assume that there is a free group action of automorphisms on a bipartite graph. If there is a perfect matching on the factor graph, then obviously there is a perfect matching on the graph. Surprisingly, the reversed is also true for amenable groups: if there is a perfect matching on the graph, there is also a perfect matching on the factor graph, i. e. a group invariant ("symmetric") perfect matching on the graph.
Ostrom and Wagner (1959) proved that if the automorphism group $G$ of a finite projective plane $π$ acts $2$-transitively on the points of $π$, then $π$ is isomorphic to the Desarguesian projective plane and $G$ is isomorphic to $\mathrm{PΓL}(3,q)$ (for some prime-power $q$). In the more general case of a finite rank $2$ irreducible spherical building, also known as a \emph{generalized polygon}, the theorem of Fong and Seitz (1973) gave a classification of the \emph{Moufang} examples. A conjecture of Kantor, made in print in 1991, says that there are only two non-classical examples of flag-tra…
We say that a group $G$ has Bergman's property (the property of universality of finite width) if for every generating set $X$ of $G$ with $X=X^{-1}$ we have that $G=X^k$ for some natural number $k.$ The property is named after George Bergman who have proved recently that the infinite symmetric groups are groups of universally finite width. The first example of an infinite group with Bergman's property is due to Shelah (1980s). Lately some other examples have been found: the automorphism groups of doubly transitive chains (Droste-Holland), the automorphism group of reals as a Borel space (Drost…
Let k be a perfect field of characteristic p > 0, and let G be a finite group. We consider the pointed G-curves over k associated by Harbater, Katz, and Gabber to faithful actions of G on k[[t]] over k. We use such "HKG G-curves" to classify the automorphisms of k[[t]] of p-power order that can be expressed by particularly explicit formulas, namely those mapping t to a power series lying in a Z/pZ Artin-Schreier extension of k(t). In addition, we give necessary and sufficient criteria to decide when an HKG G-curve with an action of a larger finite group J is also an HKG J-curve.
In this paper, we consider the problem on the existence of perfect state transfer(PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and det…
Let F_n be the free group of rank n and let Aut^+(F_n) be its special automorphism group. For an epimorphism pi : F_n -> G of the free group F_n onto a finite group G we call Gamma^+(G,pi) = {f in Aut^+(F_n) | pi*f = pi} the standard congruence subgroup of Aut^+(F_n) associated to G and pi. In the case n = 2 we fully describe the abelianization of Gamma^+(G,pi) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Gamma^+(G,pi) < Aut^+(F_2) has infinite abelianization.
The connection between classical model theoretical types (MT-types) and logically-geometrical types (LG-types) introduced by B. Plotkin is considered. It is proved that MT-types of two $n$-tuples in two universal algebras coincide if and only if their LG-types coincide. An algebra $H$ is called logically perfect if for every two $n$-tuples in $H$ whose types coincide, one can be sent to another by means of an automorphism of this algebra. Some sufficient condition for logically perfectness of free finitely generated algebras is given which helps to prove that finitely generated free Abelian gr…
This work examines the commutator structure of some closed subgroups of the wild group of automorphisms of a local field with perfect residue field, a group we call $\Cal J.$ In particular, we establish a new approach to evaluating commutators in $\Cal J$ and using this method investigate the normal subgroup structure of some classes of index subgroups of $\Cal J$ as introduced by Klopsch. We provide new proofs of Fesenko\rq s results that lead to a proof that the torsion free group $T =\{t+\sum_{k\geq 1} a_kt^{qk+1}: a_k \in \Bbb F_p\}$ is hereditarily just infinite, and by extending his work…
Let $G$ be a group. Two elements $x,y \in G$ are said to be in the same $z$-class if their centralizers in $G$ are conjugate within $G$. Consider $\mathbb F$ a perfect field of characteristic $\neq 2$, which has a non-trivial Galois automorphism of order $2$. Further, suppose that the fixed field $\mathbb F_0$ has the property that it has only finitely many field extensions of any finite degree. In this paper, we prove that the number of $z$-classes in the unitary group over such fields is finite. Further, we count the number of $z$-classes in the finite unitary group $U_n(q)$, and prove that …
In this thesis, we develop algorithms similar to the Gaussian elimination algorithm in symplectic and split orthogonal similitude groups. As an application to this algorithm, we compute the spinor norm for split orthogonal groups. Also, we get similitude character for symplectic and split orthogonal similitude groups, as a byproduct of our algorithms. Consider a perfect field k with odd characteristics, which has a non-trivial Galois automorphism of order 2. Further, suppose that the fixed field k_0 has the property that there are only finitely many field extensions of any finite degree. In th…
A loop is automorphic if all its inner mappings are automorphisms. We construct a large family of automorphic loops as follows. Let $R$ be a commutative ring, $V$ an $R$-module, $E=\mathrm{End}_R(V)$ the ring of $R$-endomorphisms of $V$, and $W$ a subgroup of $(E,+)$ such that $ab=ba$ for every $a$, $b\in W$ and $1+a$ is invertible for every $a\in W$. Then $Q_{R,V}(W)$ defined on $W\times V$ by $(a,u)(b,v) = (a+b,u(1+b)+v(1-a))$ is an automorphic loop. A special case occurs when $R=k<K=V$ is a field extension and $W$ is a $k$-subspace of $K$ such that $k1\cap W = 0$, naturally embedded into $\…
We show that every locally finite bipartite Borel graph satisfying a strengthening of Hall's condition has a Borel perfect matching on some comeager invariant Borel set. We apply this to show that if a group acting by Borel automorphisms on a Polish space has a paradoxical decomposition, then it admits a paradoxical decomposition using pieces having the Baire property. This strengthens a theorem of Dougherty and Foreman who showed that there is a paradoxical decomposition of the unit ball in $\mathbb{R}^3$ using Baire measurable pieces. We also obtain a Baire category solution to the dynamical…
Let R be a semi-local regular ring containing an infinite perfect field, and let K be the field of fractions of R. Let H be a simple algebraic group of type F_4 over R such that H_K is the automorphism group of a 27-dimensional Jordan algebra which is a first Tits construction. If char K is not 2, this means precisely that the f_3 invariant of H_K is trivial. We prove that if an H-torsor is rationally trivial, then it is trivial over R. This result is a particular case of the Grothendieck-Serre conjecture. It continues the recent series of papers by I. Panin, N.Vavilov and the authors, and com…
