Extending topologies from subgroups to groups. (English) Zbl 0625.22002

Given a group G, a proper subgroup H and a topological group topology \(\tau\) for H, the authors raise this question: Is there a topological group topology \({\mathcal T}\) for G such that \(\tau\) is a local \({\mathcal T}\)- base at e ? Among the results proved here are these: (a) The answer is “Yes” iff the three families \({\mathcal T}\), \(\{\) xU: \(x\in G\), \(U\in \tau \}\), and \(\{\) Ux: \(x\in G\), \(U\in \tau \}\) are equal; (b) for H normal in G, the answer is “Yes” iff each conjugation map \(C_ g: H\to H\) (defined by \(C_ g(x)=gxg^{-1})\) with \(g\in G\) is a \(\tau\)- homeomorphism; and (c) there exist G with normal H for which the answer is “No”. In addition, the authors extend the class \({\mathcal G}\) of groups for which it is known that every \(G\in {\mathcal G}\) admits a non-trivial topological group topology.
Reviewer: W.W.Comfort


22A05 Structure of general topological groups
54H15 Transformation groups and semigroups (topological aspects)