R. Guralnick, G Roehrle
The goal of this note is to classify the weakly closed unipotent subgroups in the split Chevalley groups. In an application we show under some mild assumptions on the characteristic that the Lie algebra of a connected simple algebraic group fails to be a so called 2F-module.
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We show that all unipotent classes in finite simple Chevalley or Steinberg groups, different from PSL_n(q) and PSp_{2n}(q), collapse (i.e. are never the support of a finite-dimensional Nichols algebra), with a possible exception on one class of involutions in PSU_n(2^m).
Let G be a finite simple group of Lie type. In this paper we study characters of G that vanish at the non-semisimple elements and whose degree is equal to the order of a maximal unipotent subgroup of G. Such characters can be viewed as a natural generalization of the Steinberg character. For groups G of small rank we also determine the characters of this degree vanishing only at the non-identity unipotent elements.
The current article continues a series of papers on decomposition of unipotents and its applications. Let $G(Φ,R)$ be a Chevalley group with a reduced irreducible root system $Φ$ over a commutative ring $R$. Fix $h\in G(Φ,R)$. Call an element $a\in G(Φ,R)$ "good", if it lies in the unipotent radical of a parabolic subgroup whereas the conjugate to $a$ by $h$ belongs to another proper parabolic subgroup (here we assume that all parabolics contain a given split maximal torus). Decomposition of unipotents is a representation of a root unipotent element as a product of "good" elements. Existence o…
We prove that finite index subgroups in S-arithmetic Chevalley groups are bounded.
We prove the centrality of the congruence kernel for the elementary subgroup of a Chevalley group G of rank >1 over an arbitrary noetherian ring R (under some minor restrictions on R if G is of type C_n or G_2).
Arithmetic subgroups of Chevalley group schemes over function fields II: Conjugacy classes of maximal unipotent subgroups
本文研究函数域上 $\mathcal{S}$-算术子群的极大幂单子群共轭类，利用 Picard 群与群秩加以参数化，并通过 Bruhat–Tits 建筑扇形稳定子刻画其结构。
We formulate and prove relative versions of several classical decompositions known in the theory of Chevalley groups over commutative rings. As an application we obtain upper estimates for the width of principal congruence subgroups in terms of several families of generators. Some of our results are new even in the absolute case and were previously studied only for groups over finite fields.
This paper describes how to use subgroups to parameterize unipotent classes in the classical algebraic group in characteristic 2. These results can be viewed as an extension of the Bala-Carter Theorem, and give a convenient way to compare unipotent classes in a group $G$ with unipotent classes of a subgroup $X$ where $G$ is exceptional and $X$ is a Levi subgroup of classical type.
Separability of unipotent-free abelian subgroups in linear groups
本文证明了在零特征域上的有限生成线性群中，无幂单阿贝尔子群具有可分性。
Lately, the following problem has attracted a lot of attention in various contexts: find the shortest factorisation $G=UU^-UU^-...U^{\pm}$ of a Chevalley group $G=G(Φ,R)$ in terms of the unipotent radical $U=U(Φ,R)$ of the standard Borel subgroup $B=B(Φ,R)$ and the unipotent radical $U^-=U^-(Φ,R)$ of the opposite Borel subgroup $B^-=B^-(Φ,R)$. So far, the record over a finite field was established in a 2010 paper by Babai, Nikolov, and Pyber, where they prove that a group of Lie type admits unitriangular factorisation $G=UU^-UU^-U$ of length 5. Their proof invokes deep analytic and combinatori…
Random unipotent Sylow subgroups of groups of Lie type of bounded rank
该研究针对有界秩有限单李型群，证明其可表为11个随机幂零Sylow子群的乘积且概率随群阶趋于无穷趋近1，给出有限线性群应用，证明不依赖有限单群分类。
Let F be a local field of positive characteristic, and let G be either a Heisenberg group over F, or a certain (nonabelian) two-dimensional unipotent group over F. If H is an arithmetic subgroup of G, we provide an explicit description of every automorphism of H. From this description, it follows that every automorphism of H virtually extends to a virtual automorphism of G.
Unipotent normal subgroups of algebraic groups
本文研究了代数群的单势正则子群的性质，并探讨了其在光滑连通仿射群中的类比关系。
Every commutator preserving bijection of the unipotent radical $Up(2n, R)$ of the Borel subgroup of the classical symplectic group of rank at least 4 over a field $F$ such that $6F=F$ is shown to be the composition of a standard automorphism of $Up(2n,R)$ and a central map. The latter is a bijection which acts as the right multiplication by a matrix from the center of $Up(2n,F)$.
We continue study of subgroups of a Chevalley group $G_P(Φ,R)$ over a ring $R$ with a root system $Φ$ and a weight lattice $P$, containing the elementary subgroup $E_P(Φ,K)$ over a subring $K$ of $R$. Recently A. Bak and A. Stepanov considered the symplectic case (i. e. the case of simply connected group of type $Φ=C_l$) in characteristic 2. In this article we extend their result for groups with arbitrary weight lattice of types $B_l$ and $C_l$. Similarly to the work of Nuzhin that handles the case of an algebraic extension $R$ of a nonperfect field $K$ of bad characteristic, we use in the des…
In studying nilpotent groups, the lower central series and other variations can be used to construct an associated $\mathbb{Z}^+$-graded Lie ring, which is a powerful method to inspect a group. Indeed, the process can be generalized substantially by introducing $\mathbb{N}^d$-graded Lie rings. We compute the adjoint refinements of the lower central series of the unipotent subgroups of the classical Chevalley groups over the field $\mathbb{Z}/p\mathbb{Z}$ of rank $d$. We prove that, for all the classical types, this characteristic filter is a series of length $Θ(d^2)$ with nearly all factors ha…
In the present paper we prove sandwich classification for the overgroups of the subsystem subgroup $E(Δ,R)$ of the Chevalley group $G(Φ,R)$ for the three types of pair $(Φ,Δ)$ (the root system and its subsystem) such that the group $G(Δ,R)$ is (up to torus) a Levi subgroup of the parabolic subgroup with abelian unipotent radical. Namely we show that for any such an overgroup $H$ there exists a unique pair of ideals $σ$ of the ring $R$ such that $E(Φ,Δ,R,σ)\le H\le N_{G(Φ,R)}(E(Φ,Δ,R,σ))$.
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 $U = \mathbf U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G = \mathbf G(q)$. In [GR}] Röhrle and the second author determined a parameterization of the conjugacy classes of $U$, for $\mathbf G$ of small rank when $q$ is a power of a good prime for $\mathbf G$. As a consequence they verified that the number $k(U)$ of conjugacy classes of $U$ is given by a polynomial in $q$ with integer coefficients. In the present paper, we consider the case when $p$ is a bad prime for $\mathbf G$. We obtain a parameterization of the conjugacy classes of $U$, when $\mathbf G$ has rank less tha…
On Normal Subgroups of Twisted Chevalley Groups over Commutative Rings
本文研究了扭曲Chevalley群在可换环上的结构，证明了初等同余子群的正规性及子群分类，同时探讨了半局部环下的群分解性质。
We prove a formula connecting the number of unipotent conjugacy classes in a maximal parabolic subgroup of a finite general linear group with the numbers of unipotent conjugacy classes in various parabolic subgroups in smaller dimensions. We generalise this formula and deduce a number of corollaries; in particular, we express the number of conjugacy classes of unitriangular matrices over a finite field in terms of the numbers of unipotent conjugacy classes in maximal parabolic subgroups over the same field. We show how the numbers of unipotent conjugacy classes in parabolic subgroups of small …
Let $U$ be a Sylow $p$-subgroup of the finite Chevalley group of type $D_4$ over the field of $q$ elements, where $q$ is a power of a prime $p$. We describe a construction of the generic character table of $U$.
Let $G$ be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic $p>0$ and let $X = {\rm PSL}_{2}(p)$ be a subgroup of $G$ containing a regular unipotent element $x$ of $G$. By a theorem of Testerman, $x$ is contained in a connected subgroup of $G$ of type $A_1$. In this paper we prove that with two exceptions, $X$ itself is contained in such a subgroup (the exceptions arise when $(G,p) = (E_6,13)$ or $(E_7,19)$). This extends earlier work of Seitz and Testerman, who established the containment under some additional conditions on $p$ and the …
We develop in this work a method to parametrize the set $\mathrm{Irr}(U)$ of irreducible characters of a Sylow $p$-subgroup $U$ of a finite Chevalley group $G(p^f)$ which is valid for arbitrary primes $p$, in particular when $p$ is a very bad prime for $G$. As an application, we parametrize $\mathrm{Irr}(U)$ when $G=\mathrm{F}_4(2^f)$.
Let $U(q)$ be a Sylow $p$-subgroup of the Chevalley groups $D_4(q)$ where $q$ is a power of a prime $p$. We describe a construction of all complex irreducible characters of $U(q)$ and obtain a classification of these irreducible characters via the root subgroups which are contained in the center of these characters. Furthermore, we show that the multiplicities of the degrees of these irreducible characters are given by polynomials in $q-1$ with nonnegative coefficients.
Given a universal elementary Chevalley-Demazure group $E_Φ^{sc}(R)$ for which its (standard) parabolic subgroups are finitely generated, we consider the problem of classifying which parabolics $P(R) \subset E_Φ^{sc}(R)$ are finitely presented. We show that, under mild assumptions, this is equivalent to the finite presentability of a suitable retract of $P$ which contains the Levi factor. If the base ring $R$ is a Dedekind domain of arithmetic type, we combine our results with well-known theorems due to Borel-Serre, Abels, Behr and Bux to give a partial classification of finitely presentable $S…
We consider semi-continuity of certain dimensions on group schemes.
Let G(q) be a finite Chevalley group, where q is a power of a good prime p, and let U(q) be a Sylow p-subgroup of G(q). Then a generalized version of a conjecture of Higman asserts that the number k(U(q)) of conjugacy classes in U(q) is given by a polynomial in q with integer coefficients. In an earlier paper, the first and the third authors developed an algorithm to calculate the values of k(U(q)). By implementing it into a computer program using GAP, they were able to calculate k(U(q)) for G of rank at most 5, thereby proving that for these cases k(U(q)) is given by a polynomial in q. In thi…
Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of $G$ in characteristic other than p to a maximal subgroup of $G_2(q)$ is still irreducible. Similar results are obtained for $^2B_2(q)$ and $^2G_2(q)$.
Unicity of A1-subgroups associated to unipotent elements in simple algebraic groups
本文研究了正特征单连通代数群中幂幺元关联的 A1-子群的共轭唯一性问题，确定了满足共轭性的极大 G-稳定子簇，从而明确了 Jacobson-Morozov 定理在正特征代数群中的适用程度。
We prove that Chevalley group over the field $F$ of zero characteristic possess $R_{\infty}$ property, if $F$ has torsion group of automorphisms or $F$ is an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. As a consequence we obtain that the twisted conjugacy class $[e]_φ$ of unit element is a subgroup of Chevalley group if and only if $φ$ is central automorphism.
Let $G$ be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank $\geq 2$, and let $R$ be a commutative ring. We analyze the linear representations $ρ\colon G(R)^+ \to GL_n (K)$ over an algebraically closed field $K$ of the elementary subgroup $G(R)^+ \subset G(R).$ Our main result is that under certain conditions, any such representation has a standard description, i.e. there exists a commutative finite-dimensional $K$-algebra $B$, a ring homomorphism $f \colon R \to B$ with Zariski-dense image, and a morphism of algebraic groups $σ\colon G(B…
Let G be a unipotent algebraic subgroup of some GL_m(C) defined over Q. We describe an algorithm for finding a finite set of generators of the subgroup G(Z) = G \cap GL_m(Z). This is based on a new proof of the result (in more general form due to Borel and Harish-Chandra) that such a finite generating set exists.
The existence of closed orbits of real algebraic groups on real algebraic varieties is established. As an application, it is shown that if G is a real reductive linear group with Iwasawa decomposition G= KAN, then every unipotent subgroup of G is conjugate to a subgroup of N.
In this article we improve the known uniform bound for subgroup growth of Chevalley groups over $\mathbf{G}(\mathbb{F}_p[[t]])$. We introduce a new parameter, the ridgeline number $v(\mathbf{G})$, and give new bounds for the subgroup growth of $\mathbf{G}(\mathbb{F}_p[[t]])$ expressed through $v(\mathbf{G})$. We achieve this by deriving a new estimate for the codimension of $[U,V]$ where $U$ and $V$ are vector subspaces in the Lie algebra of $\mathbf{G}$.
In earlier work, the first author outlined an algorithm for calculating a parametrization of the conjugacy classes in a Sylow $p$-subgroup $U(q)$ of a finite Chevalley group $G(q)$, valid when $q$ is a power of a good prime for $G(q)$. In this paper we develop this algorithm and discuss an implementation in the computer algebra language {\sf GAP}. Using the resulting computer program we are able to calculate the parametrization of the conjugacy classes in $U(q)$, when $G(q)$ is of rank at most 6. In these cases, we observe that the number of conjugacy classes of $U(q)$ is given by a polynomial…
The current paper is an addition to the previous paper by author, where the overgroup lattice of the elementary subsystem subgroup $E(Δ,R)$ of the Chevalley group $G(Φ,R)$ for a large enough root subsystem $Δ$ was studied. Now we study the connection between the elementary subgroup $\hat{E}(σ)$ given by the net of ideals of the ring $R$ and the stabilizer $S(σ)$ of the corresponding Lie subalgebra of the Chevalley algebra. In particular, we prove that under a certain condition the subgroup $\hat{E}(σ)$ is normal in $S(σ)$, and we also study some properties of the corresponding quotient group.
One of the central tools in the classification of simple algebraic groups is the distinction between semisimple subgroups and unipotent subgroups. It is not a priori clear how to make this distinction for torsion-free subgroups of a group of finite Morley rank. We exploit the ``graded'' notion of 0-unipotence to develop a Sylow theory for torsion-free subgroups of a solvable group of finite Morley rank. This Sylow theory provides a robust alternative to the usual theory of Carter subgroups, and will be used in the analysis of intersections of Borel subgroups in minimal simple groups.
Automorphisms of twisted Chevalley groups of type ${}^2E_6$ over local rings
本文分类了带对合且2可逆的交换局部环上，两种等义类型的${}^2E_6$型扭Chevalley群的抽象自同构，证明其均为标准型，明确了伴随与单连通情形下自同构的结构，通过基变换替换等步骤完成分类。
Let $\F_q$ be a field of characteristic $p$ with $q$ elements. It is known that the degrees of the irreducible characters of the Sylow $p$-subgroup of $GL_n(\F_q)$ are powers of $q$ by Issacs. On the other hand Sangroniz showed that this is true for a Sylow $p$-subgroup of a classical group defined over $\F_q$ if and only if $p$ is odd. For the classical groups of Lie type $B$, $C$ and $D$ the only bad prime is 2. For the exceptional groups there are others. In this paper we construct irreducible characters for the Sylow $p$-subgroups of the Chevalley groups $D_4(q)$ with $q=2^f$ of degree $q^…
The unipotent groups are an important class of algebraic groups. We show that techniques used to compute with finitely generated nilpotent groups carry over to unipotent groups. We concentrate particularly on the maximal unipotent subgroup of a split reductive group and show how this improves computation in the reductive group itself.
We investigate positive-dimensional closed reductive subgroups of almost simple algebraic groups containing a regular unipotent element. Our main result states that such subgroups do not lie inside proper parabolic subgroups unless possibly when their connected component is a torus. This extends the earlier result of Testerman and Zalesski treating connected reductive subgroups.
Given a Chevalley group ${\mathbf G}(q)$ and a parabolic subgroup $P\subset {\mathbf G}(q)$, we prove that for any set $A$ there is a certain growth of $A$ relatively to $P$, namely, either $AP$ or $PA$ is much larger than $A$. Also, we study a question about intersection of $A^n$ with parabolic subgroups $P$ for large $n$. We apply our method to obtain some results on a modular form of Zaremba's conjecture from the theory of continued fractions and make the first step towards Hensley's conjecture about some Cantor sets with Hausdorff dimension greater than $1/2$.
The paper studies lower bounds for the dimensions of projective indecomposable modules for Chevalley groups G in defining characteristic p. The main result extending earlier one by Malle and Weigel (2008) determines the modules in question of dimension equal to the order of a Sylow p-subgroup of G. We also substantially generalize a result by Ballard (1978) on lower bounds for the dimensions of projective indecomposable modules and find lower bounds in some cases where Ballard's bounds are vacuous.
Lie type quotients of the maximal unipotent subgroup of Kac-Moody groups of type $\mathrm{HB}_{2}^{(2)}$
本文构造了秩无界的 Lie 型有限单群无限族作为类型 $\mathrm{HB}_{2}^{(2)}$ 的 Kac-Moody 群极大幂幺子群的商，并据此首次构建出由无界秩 Lie 型群生成的高维扩展图无限族。
Some Properties of Twisted Chevalley Groups
本论文研究交换环上扭Chevalley群的结构性质，探讨了相对初等子群的正规性、被初等子群正规化的子群分类，以及在环扩张中正规化子的重合性，证明了伴随型群的正规化子恰好等于原群。
Let $q$ be a power of a prime $p$, let $G$ be a finite Chevalley group over $\mathbb{F}_q$ and let $U$ be a Sylow $p$-subgroup of $G$; we assume that $p$ is not a very bad prime for $G$. We explain a procedure of reduction of irreducible complex characters of $U$, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of $U$ along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when $G$ is of type $\mathrm{F}_4$, where we observe that the parametrization is "uniform" over goo…
Arithmetic subgroups of Chevalley group schemes over function fields I: quotients of the Bruhat-Tits building by $\{P\}$-arithmetic subgroups
本文研究函数域上 Chevalley 群概形的 A-点算术子群在 Bruhat-Tits 建筑上的作用，证明其轨道空间可描述为闭连通 CW-复形与由 Picard 群和群秩参数化的扇形腔室的粘合，并刻画了可嵌入轨道空间的子扇形面。
We show that all classes that are neither semisimple nor unipotent in finite simple Chevalley or Steinberg groups different from $PSL_n(q)$ collapse (i.e. are never the support of a finite-dimensional Nichols algebra). As a consequence, we prove that the only finite-dimensional pointed Hopf algebra whose group of group-like elements is $PSp_{2n}(q)$, $PΩ^+_{4n}(q)$, $PΩ^-_{4n}(q)$, $^3D_4(q)$, $E_7(q)$, $E_8(q)$, $F_4(q)$, or $G_2(q)$ with q even is the group algebra.
We parametrize the set of irreducible characters of the Sylow $p$-subgroups of the Chevalley groups $\mathrm{D}_6(q)$ and $\mathrm{E}_6(q)$, for an arbitrary power $q$ of any prime $p$. In particular, we establish that the parametrization is uniform for $p \ge 3$ in type $\mathrm{D}_6$ and for $p \ge 5$ in type $\mathrm{E}_6$, while the prime $2$ in type $\mathrm{D}_6$ and the primes $2,$ $3$ in type $\mathrm{E}_6$ yield character degrees of the form $q^m/p^i$ which force a departure from the generic situations. Also for the first time in our analysis we see a family of irreducible characters …
In the 1960's Noboru Iwahori and Hideya Matsumoto, Eiichi Abe and Kazuo Suzuki, and Michael Stein discovered that Chevalley groups $G=G(Φ,R)$ over a semilocal ring admit remarkable Gauss decomposition $G=TUU^-U$, where $T=T(Φ,R)$ is a split maximal torus, whereas $U=U(Φ,R)$ and $U^-=U^-(Φ,R)$ are unipotent radicals of two opposite Borel subgroups $B=B(Φ,R)$ and $B^-=B^-(Φ,R)$ containing $T$. It follows from the classical work of Hyman Bass and Michael Stein that for classical groups Gauss decomposition holds under weaker assumptions such as $\sr(R)=1$ or $\asr(R)=1$. Later the second author no…
Sha-rigidity of Chevalley groups over local rings
本文研究局部环上 Chevalley 群的刚性问题，证明在特定根系及可逆元条件下，其所有局部内自同态均为内自同态，从而确立这些群的 Sha 刚性。
Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of $U(q)$ when $G(q)$ is of type $\mathrm{G}_2$. This is uniform for primes $p \ge 5$, while the bad primes $p=2$ and $p=3$ have to be considered separately. We then use this result and the contribution of several authors to show a general result, namely that if $G(q)$ is any finite Chevalley group with $p$ a bad prime, then there exists a character $χ\in \tex…
Overgroups of regular unipotent elements in reductive groups
本文研究约化线性代数群中包含正则幂单元素的约化子群，证明在适当条件下这些子群是Serre意义下的G-不可约的，推广了已有结果并涵盖李代数和有限李型群，证明简洁统一。
Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \u its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \u* of \u. This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which h…
Let $G$ be a simple algebraic group over an algebraically closed field $k$ of characteristic $p$. The classification of the conjugacy classes of unipotent elements of $G(k)$ and nilpotent orbits of $G$ on $\operatorname{Lie}(G)$ is well-established. One knows there are representatives of every unipotent class as a product of root group elements and every nilpotent orbit as a sum of root elements. We give explicit representatives in terms of a Chevalley basis for the eminent classes. A unipotent (resp. nilpotent) element is said to be eminent if it is not contained in any subsystem subgroup (re…
In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chevalley groups. Namely, let $Φ$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(Φ,R)$ of type $Φ$ over $R$. The unrelativised elementary subgroup $E(Φ,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_α(ξ)$, $α\inΦ$, $ξ\in I$, of level $I$. Obviously, in general $E(Φ,I)$ has n…
Let $q$ be a power of a prime $p$. Let $P$ be a parabolic subgroup of the general linear group $\GL_n(q)$ that is the stabilizer of a flag in $\FF_q^n$ of length at most 5, and let $U = O_p(P)$. In this note we prove that, as a function of $q$, the number $k(U)$ of conjugacy classes of $U$ is a polynomial in $q$ with integer coefficients.
Describing the conjugacy classes of the unipotent upper triangular groups $\mathrm{UT}_{n}(\mathbb{F}_{q})$ uniformly (for all or many values of $n$ and $q$) is a nearly impossible task. This paper takes on the related problem of describing the normal subgroups of $\mathrm{UT}_{n}(\mathbb{F}_{q})$. For $q$ a prime, a bijection will be established between these subgroups and pairs of combinatorial objects with labels from $\mathbb{F}_{q}^{\times}$. Each pair comprises a loopless binary matroid and a tight splice, an apparently new kind of combinatorial object which interpolates between nonnesti…
Let K be a field and g(K) a Chevalley group (scheme) over K. Let (B,N) be the standard spherical BN-pair in g(K), with T=B\cap N and Weyl group W=N/T. We prove that there exist non-trivial elements w\in W such that all representatives of w in N have finite order. This allows us to exhibit examples of subgroups of g(Q_p) that act Weyl transitively but not strongly transitively on the affine building Delta associated with g(Q_p). Such examples were previously known only in the case when g(Q_p)=SL_2(Q_p) and Delta is a tree.
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