Automorphisms of Chevalley groups of different types over commutative rings

Abstract

In this paper we prove that every automorphism of (elementary) adjoint Chevalley group with root system of rank > 1 over a commutative ring (with 1/2 for the systems A 2 , F 4 , B l , C l ; with 1/2 and 1/3 for the system G 2 ) is standard, i. e., it is a composition of ring, inner, central and graph automorphisms.

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

“…A similar result for local rings without 1/2 was obtained in [5]. In [6], similar statements for the root system F 4 were proved, and in [7] all previous results with the help of localization method were generalized for the case of adjoint Chevalley groups over arbitrary commutative rings.…”

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

“…Theorem 1 for the root systems B 2 and G 2 was obtained in [5], but in this paper for the root system G 2 1/2 and 1/3 are obtained to be in the ring. In [10], E. I. Bunina proved similar theorems for the root system F 4 ; in [11], all previous results (with the help of localization method) were generalized for the case of adjoint Chevalley groups over arbitrary commutative rings (with corresponding conditions of existence of 1/2 or 1/3).…”

Twisted conjugacy in classical groups over certain domains of Characteristic $p>0$

“…We assume that the root system Φ is irreducible of classical type, that is, we assume that Φ is A l for l ≥ 1, B l for l ≥ 2, C l for l ≥ 3 or D l for l ≥ 4. The automorphisms of G(Φ, R) have been discussed in various papers but our reference is [7] since it is one of the recent works on this topic and also because it gives a uniform description applicable to all types valid for every R.…”

“…Roughly speaking, the proof of 1.1 involves three major steps which are as follows: (i) to have a nice description of the automorphisms of the groups of our consideration, which is given by E. Bunina [7] and is recalled in §3 of this paper, (ii) find a convenient representative automorphism φ within its outer automorphism class and use Lemma 2.1, (iii) come up with an infinite sequence x k , k ∈ N of elements in the group which lie in distinct φ-twisted conjugacy classes and thus conclude the R ∞ -property of the group.…”