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On full subgroups of Chevalley groups. (English) Zbl 0598.20051
Let G be a split algebraic absolutely almost simple group defined over a field k. For a split maximal k-torus T of G, let $\Sigma =\Sigma (G,T)$ denote the root system of G with respect to T, $\{x\sb{\epsilon}$; $\epsilon\in \Sigma \}\sp a $system of isomorphisms, normalized as usual, from the additive groups onto the root subgroups of G. The author defines that a subgroup H of G(k) is full if for every $g\in G(k)$ and $\epsilon\in \Sigma$, there exists a non-zero element $c=c(g,\epsilon)\in k$ such that $g\sp{-1}x(c)g\in H$, and a subset R of k is full if for every $y\in k$ there exists a non-zero $r\in R$ such that yr$\in R$. For a subset R of k, let $G\sp E(R)$ be the subgroup of G(k) generated by all $x\sb{\epsilon}(a)$, where $\epsilon\in \Sigma$ and $a\in R$. The author proves among others the following, here a subring means that k is its field of fractions and not required to have identity and rank G$>1.$ For every subring R of k, $G\sp E(R)$ is a full subgroup of G(k); Every full subgroup H of G(k) contains $G\sp E(A)$ for some full subring A of k with the exception of the case when G is of type $C\sb n$ (n$\ge 2)$, $ch(k)=2$ and dimension of k over $k\sp 2$ is uncountable. In the exceptional case, it is shown that not every full subgroup contains $G\sp E(A)$ for a full subring A; If H is a full subgroup of G(k), and $g\sb 1,...,g\sb n\in G(k)$, then $\cap\sb{i}g\sb iHg\sb i\sp{-1}$ is a full subgroup of G(k); Assume that k does not consist of 2 elements when G is of type $B\sb 2$ or $G\sb 2$. If H is full and M is a subgroup of G(k) normalized by H, then $H\cap M$ is full or M lies in the center of G. Further the author discusses some types of subgroups similar to ”tableau”, ”carpet” or ”net” groups considered by Riehm, James, Borevich, Vavilov and others, but the ideals are replaced by more general modules.
Reviewer: E.Abe

20G15Linear algebraic groups over arbitrary fields
20E07Subgroup theorems; subgroup growth
Full Text: DOI
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