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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

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