Reduction theorems for groups of matrices
Let \(G\) be either a compact Lie group or a linear algebraic group over an algebraically closed field \(k\). The authors prove results of the following kind: \(G\) has a given property if and only if every finite subgroup has that property. Theorem 4 asserts that this is true for the properties of being solvable, nilpotent or abelian provided that in the latter two cases; the unipotent radical, \(G^0\), of \(G\) is trivial when \(\text{char}(k) = 0\). Simple examples show that the last proviso is necessary and that the theorem does not extend to non-compact Lie groups. The essential step in proving Theorem 4 for a linear algebraic group \(G\) is to show that, when \(G^0\) is a torus, \(G\) contains a locally finite, Zariski dense subgroup. Theorem 7 is a refinement of this result in the case where \(\text{char}(k)= p > 0\): here, the conclusion holds without restriction on \(G^0\).
scientific article; zbMATH DE number 2081927