Free product subgroups between Chevalley groups G(φ, F) and G(φ, F[t])

JOURNAL OF ALGEBRA

卷:324

Article

Stepanov, Alexei^1,2 

[1] St Petersburg Electrotech Univ, St Petersburg 197376, Russia

[2] Govt Coll Univ, Abdus Salam Sch Math Sci, Lahore, Pakistan

关键词: Chevalley groups;    Lattice of subgroups;    Free product;   

DOI  :  10.1016/j.jalgebra.2010.06.015

来源: Elsevier

【 摘 要 】

We investigate subgroups of a Chevalley group G = G(phi, A) over a ring A, containing its elementary subgroup E = E(phi, F) over a subring F subset of A. Assume that the root system phi is simply laced and A = F[t] is a polynomial ring. We show that if G is of adjoint type, then there exists an element g is an element of E(phi, A) such that < g, E(phi, F)> = < g > * E(phi, F), where (X) denotes the subgroup, generated by a set X. and * stands for the free product. It follows that under the above assumptions the lattice L = L(E, G) is not standard. Moreover, combining the above result with theorems of Nuzhin and the author one obtains a necessary and sufficient condition for L to be standard provided that A and F are fields of characteristic not 2 and phi not equal G(2). (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

Free