The Frattini subgroup of a Lie group and the topological rank of a Lie group (Q6142853)

In the paper under review, the authors define the Frattini subgroup \(\Phi(G)\) of a topological group \(G\) as the set of non-generators (\(g \in \Phi(G)\) if and only if every subset \(X\) of \(G\) such that \(\overline{ \langle X \cup \{g\} \rangle}=G\) is such that \(\overline{\langle X \rangle}=G\). For other equivalent conditions see Definition 2.1). A direct consequence of the definition is Theorem 3.5: Let \(G\) be a topological group and let \(Z\) be its center. Then \(Z\cap G'\) is contained in \(\Phi(G)\). If \(G\) is a connected Lie group, then, in general, \(\Phi(G)\) is not connected and not closed, so it is natural to consider its connected component \(\Phi^{0}(G)\). From Theorem 3.5 the following results follow. Theorem 4.1: Let \(G\) be a connected nilpotent topological group. Then \(\overline{G'}\) is contained in \(\Phi(G)\), hence \(\Phi^{0}(G)=\overline{G'}\). Theorem 4.3: Let \(G\) be a connected solvable Lie group. Then \(Z\cap \overline{G'}\) is connected and hence contained in \(\Phi^{0}(G)\). Another interesting result is Theorem 5.2: Let \(G\) be a connected Lie group and let \(R\) be its radical. Then \(\Phi(R) \subset \Phi(G)\). The authors develop the Frattini subgroup theory in topological groups in close analogy with that of finite groups. An important criterion for nilpotency is given in Theorem 5.15: Let \(G\) be a connected Lie group and let \(N\) be a normal subgroup of \(G\). Then \(N\) is nilpotent if (and only if) the image of \(N\) in \(G/\Phi(G)\) is nilpotent. In particular, \(\Phi(G)\) is nilpotent and \(G\) is nilpotent if (and only if) \(G/\Phi(G)\) is nilpotent.