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

Abstract

In this paper, we prove that every element of the linear group GL14(R) normalizing the Chevalley group of type G2 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 G2 over a local ring without 1/2 is a composition of a ring and an inner automorphisms.

Automorphisms of Chevalley groups of different types over commutative rings

“…In the case of local rings without 1/2 and the root system A 2 there is a non-standard automorphism of corresponding Chevalley group too (see [31] and [32]). For the root system G 2 non-invertible 3 is a significant obstacle for standardity of automorphisms, since even for fields of characteristic 3 a non-standard automorphism exists (see [35]), but as we see in the paper [11], non-invertibility of 2 does not interfere with standardity of automorphisms for fields and for local rings.…”

“…In the series of papers [19], [16], [18], [20], [24] the similar methods made it possible to obtain the standardity of all automorphisms of Chevalley groups G(Φ, R) where Φ = F 4 , B l , l 3, R is a local ring and 1/2 ∈ R, or Φ = G 2 and 1/2, 1/3 ∈ R. The same is true for Φ = A l , D l E l , G 2 , l 2, R is a local ring and 1/2 / ∈ R. As we already mentioned the case C l (symplectic linear groups and projective symplectic linear groups) was considered in the papers of Petechuk and Golubchik-Mikhalev (even for non-commutative rings).…”