Chevalley Group Of Type Research Articles

Chevalley groups of types $B_n$, $C_n$, $D_n$ over certain fields do not possess the $R_{\infty}$-property

Let $F$ be an algebraically closed field of zero characteristic. If the transcendence degree of $F$ over $\mathbb{Q}$ is finite, then all Chevalley groups over $F$ are known to possess the $R_{\infty}$-property. If the transcendence degree of $F$ over $\mathbb{Q}$ is infinite, then Chevalley groups of type $A_n$ over $F$ do not possess the $R_{\infty}$-property. In the present paper we consider Chevalley groups of classical series $B_n$, $C_n$, $D_n$ over $F$ in the case when the transcendence degree of $F$ over $\mathbb{Q}$ is infinite, and prove that such groups do not possess the $R_{\infty}$-property.

The Width of Extraspecial Unipotent Radical with Respect to a Set of Root Elements

Let G = G(Φ,K) be a Chevalley group of type Φ over a field K, where Φ is a simply laced root system. By studying the extraspecial unipotent radical of G, it is proved that any its element is a product of at most three root elements. Moreover, it is shown that up to conjugation by an element of the Levi subgroup, any element of the radical is the product of six elementary root elements.

Normalizers of Chevalley Groups of Type G 2 Over Local Rings Without 1/2

In this paper, we prove that every element of the linear group GL14(R) normalizing the Chevalley group of type G 2 over a commutative local ring R without 1/2 belongs to this group up to some multiplier. This allows us to improve our classification of automorphisms of these Chevalley groups showing that an automorphism-conjugation can be replaced by an inner automorphism. Therefore, it is proved that every automorphism of a Chevalley group of type G 2 over a local ring without 1/2 is a composition of a ring and an inner automorphisms.

Intermediate subgroups in the chevalley groups of type B l , C l , F 4, and G 2 over the nonperfect fields of characteristic 2 and 3

We describe the intermediate subgroups in the Chevalley groups of type B l , C l , F 4, and G 2 over various fields of characteristic 2 and 3 in the case that the larger field is an algebraic extension of the smaller nonperfect field.

Some more exceptional numerology

The paper deals with some additional details concerning the parametrization of the highest Weyl orbit of equations on the highest weight orbit in the adjoint representations of Chevalley groups of types E 7 and E 8 , as given in the author’s paper “Numerology of square equations.” Bibliography: 25 titles.

On conjugacy in a Chevalley group of large Abelian subgroups of the unipotent subgroup

Let U be the unipotent subgroup of a Chevalley group over a finite field. The well-known problem about describing the set of “large” (of maximal order) Abelian subgroups in U of exceptional type is investigated. The description of normal large Abelian subgroups in U was established earlier. It is proved that each large Abelian subgroup from U is conjugate in the Chevalley group of type F 4 over a finite field of characteristic not equal to 2 to a normal subgroup in U. It is shown that for the groups U of type G 2 and 3 D 4 a similar conclusion is not true.

Automorphisms of Chevalley groups of type B l over local rings with 1/2

In this paper, we prove that every automorphism of a Chevalley group of type Bl, l ≥ 2, over a commutative local ring with 1/2 is standard, i.e., it is a composition of ring, inner, and central automorphisms.

Identity with constants in a Chevalley group of type $\mathrm F_{4}$

Identity with constants in a Chevalley group of type $\mathrm F_{4}$

Irreducible restrictions of Brauer characters of the Chevalley group [formula omitted] to its proper subgroups

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 B 2 2 ( q ) and G 2 2 ( q ) .

The 2-Modular Reduction of the Steinberg Representation of a Finite Chevalley Group of Type C n

The 2-Modular Reduction of the Steinberg Representation of a Finite Chevalley Group of Type C n

Pairs of Short Root Subgroups in the Chevalley Group of Type G2

The paper is devoted to a description of the pairs of unipotent short root subgroups in the Chevalley group of type G 2 over a field of characteristic different from 2. Namely, the subgroups generated by a pair of short root subgroups are described, and the orbits of the Chevalley group, which acts by simultaneous conjugation on such pairs, are classified. Most of the calculations are valid for fields of characteristic 2. Bibliography: 14 titles.

K1 of Chevalley groups are nilpotent

Let Φ be a reduced irreducible root system and R be a commutative ring. Further, let G( Φ, R) be a Chevalley group of type Φ over R and E( Φ, R) be its elementary subgroup. We prove that if the rank of Φ is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G( Φ, R)/ E( Φ, R) is nilpotent by abelian. In particular, when G( Φ, R) is simply connected the quotient K 1( Φ, R)= G( Φ, R)/ E( Φ, R) is nilpotent. This result was previously established by Bak for the series A 1 and by Hazrat for C 1 and D 1. As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

PRODUCTS OF INVOLUTIONS IN THE FINITE CHEVALLEY GROUPS OF TYPE

ABSTRACT Let be a finite field of odd characteristic and let be a Chevalley group of type We find sufficient conditions for an element in to be a product of two or three involutions.

A NEW TYPE OF PRONORMAL SUBGROUPS IN CHEVALLEY GROUPS

Let L be a simple Lie algebra with irreducible root system Φ having roots of different length, F be a field of characteristic different from 2, G=L(F) be a Chevalley group of type L over F. Denote by Φl the set of all long roots in Φ. Set Gl= . It is a subgroup of G generated by all the long root subgroups. This paper determines the pronormality of Gl in G when L is not of type G2.

The Brauer trees of the exceptional Chevalley groups of types F 4 and 2 E 6

We determine the Brauer trees of the unipotent characters of the (twisted) Chevalley groups F 4 (q) and 2 E 6 (q) with one exception. The unipotent character tables of these groups are known to a large extent. This information is used in an essential way in the proofs of our results.

Generation of Certain Matrix Groups by Three Involutions, Two of Which Commute

Ž . We say that a group is 2, 2 = 2 -generated if it can be generated by three involutions, two of which commute. The problem of determining Ž . which finite simple groups are 2, 2 = 2 -generated was posed by Mazurov w x in 1980 in the Kourovka notebook 3 . An answer to this problem, for some classes of finite simple groups, was given by Ya. N. Nuzhin, namely for w x Chevalley groups of rank 1 in 4 , for Chevalley groups over a field of w x characteristic 2 in 5 , and for the alternating groups and Chevalley groups w x of type A in 6 . In this paper we consider the problem in the more n general context of matrix groups over arbitrary, finitely generated, commutative rings. As a special case of our results, stated in Theorems A, B, and Ž . C below, we obtain that most finite classical groups are 2, 2 = 2 generated, for sufficiently large rank. These groups include Chevalley groups of type B , C , and D in odd characteristic. Our results are n n n constructive, in the sense that we actually define three generating involutions, two of which commute. The choice of groups to include, and the

ON A SPECIAL LOOP, THE DICKSON FORM, AND THE LATTICE CONNECTED WITH $ O_7(3)$

A special commutative Moufang loop of order is described. With the help of this loop, a trilinear Dickson form is constructed whose automorphism group is a Chevalley group of type . Next, with the help of , a 27-dimensional representation is constructed for over , . This makes it possible to prove anew the embedding . A similar construction concerning the embedding is described.

Elements of Order Coxeter Number +1 in Chevalley Groups

Following the notation and the definitions in [1], let L(K) be the Chevalley group of type L over a field K, W the Weyl group of L and h the Coxeter number, i.e., the order of Coxeter elements of W. In a letter to the author, John McKay asked the following question: If h + 1 is a prime, is there an element of order h + 1 in L(C)? In this note we give an affirmative answer to this question by constructing an element of order h + 1 (prime or otherwise) in the subgroup Lz = 〈xτ(1)|r ∈ Φ〉 of L(K), for any K.Our problem has an immediate solution when L = An. In this case h = n + 1 and the (n + l) × (n + l) matrixhas order 2(h + 1) in SLn+1(K). This seemingly trivial solution turns out to be a prototype of general solutions in the following sense.

The second degree cohomology of finite orthogonal groups, II

Let V be a finite-dimensional vector space over GF(q), where q is odd. Suppose that dim V= 2m + 1, m = 1,2,..., and that V is equipped with a nondegenerate quadratic form Q. Let R(2m + 1, q)-or just Sa, when m and q are clear from the context--denote the commutator subgroup of the general orthogonal group of this quadratic form; that is, l2(2m + 1, q) is a Chevalley group of type B(m, q). In this paper we compute the second degree cohomology of n on its standard module V. In the smallest case (dim V= 3), H*(fi, V) is zero for q = 3, and it is of dimension 1 over GF(q), otherwise [lo]. Hence we will assume from now on that dim Va 5. Our main result is that, under these assumptions, H*(J2, V) is always zero, except for the group n(7,3), where we obtain an upper bound of 1 for the dimension of the second degree cohomology group over GF(3). The group H*(0(7, 3), V) is already known to be nonzero [5], so that it constitutes the only exception. The general outline of the proof is to calculate H*(G, V), where G is the maximal parabolic subgroup of R which is the stabilizer of a singular point. Since the restriction map from n to G induces an injection on the cohomology level, H*(R, V) will be trivial provided that H*(G, V) is. Except for the cases of q = 3, dim V= 5, 7 or 9, it is fairly straightforward to obtain the result that H’(G, V) is zero, by studying certain exact cohomology sequences obtained from the E,-terms of the Hochschild-Serre spectral sequence. In the two cases of q = 3, dim V= 5 or 9, where H*(G, V) is nonzero, we obtain explicit formulas for the nontrivial cocycle classes on the 3Sylow subgroup of the orthogonal group which is contained in the maximal parabolic subgroup G. We then show that these cocycle classes are not stable ones, hence in these two cases, H*(R, V) must be trivial again. 88 0021-8693/80/110088-22$02.00/O

Large abelian subgroups of Chevalley groups

AbstractFor any group S let Ab(S) = {A∣A is an abelian subgroup of S of maximal order}. Let G be a Chevalley group of type An, Bn, Cn, or Dn over a finite field of characteristic p and let. In this paper Ab(U) is determined for all such groups.