Subgroups generated by root-elements of groups generated by \(k\)-root subgroups
The author continues his study of groups \(G\) generated by a set \(\Sigma\) of so called \(k\)-root subgroups. For definitions see the author's papers [Invent. Math. 100, 167-206 (1990; Zbl 0697.20018) and ibid. 106, 575-666 (1991; Zbl 0794.20057)]. There the groups in question are classified. Under a simplicity condition essentially simple algebraic groups with \(\Sigma\) the class of centers of the long root subgroups come out. In the definition of \(k\)-root subgroups for a group \(G\) one of the conditions is: (*) \(G\) satisfies the maximality condition for \(\Sigma\)-subgroups. One of the purposes of the current paper is to show that all of the groups appearing in the main theorems of the papers quoted above really satisfy (*). Furthermore the author shows that if \(G\) is either a Chevalley group over a field \(k\), and \(\Sigma\) the class of long root subgroups of \(G\), or a quasisimple group generated by a class \(\Sigma\) of \(k\)-root groups satisfying (*), and \(U\) is a subgroup of \(G\) such that for \(\Sigma^ o = \{C \cap U \mid C \in \Sigma,\;C\cap U \neq 1\}\) the following hold: (1) \(U = \langle \Sigma^ o\rangle\); (2) \(U\) contains no nontrivial nilpotent normal subgroup \(N\) generated by \(N \cap \Sigma^ o\), (3) \(\Sigma^ o \neq \Sigma^ 1 \cup \Sigma^ 2\), \(\Sigma_ 1 \cap \Sigma_ 2 = \emptyset \neq \Sigma_ i\), and \([E^ o,F^ o] = 1\) for \(E^ o \in \Sigma_ 1\), \(F^ o \in \Sigma^ 2\). (4) If \(A \cap U\), \(B \cap U \in \Sigma^ o\) and \(\langle A,B \rangle \cong (P)SL_ 2(k)\), then \(\langle A \cap U, B \cap U\rangle \cong (P)SL_ 2(s)\), for a field \(s\) depending on \(A\) and \(B\). Then there is a subfield \(l\) of \(k\) such that \(\Sigma^ o\) is a class of \(l\)-root subgroups of \(U\) satisfying a weaker condition than (*) (unipotent subsets of \(\Sigma^ o\) generate nilpotent subgroups). In addition to this result there are presented a lot more interesting properties of groups generated by \(k\)-root groups.
groups generated by \(k\)-root subgroups
maximality condition for \(\Sigma\)-subgroups
scientific article; zbMATH DE number 569270