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Suppose G is a topological group containing a (closed) topological copy ofthe Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normalk-space) then every closed metrizable subset in $G$ is locally compact.Applying this result to topological groups whose underlying topological spacecan be written as a direct limit of a sequence of closed metrizable subsets, weget that every such a group either is metrizable or is homeomorphic to theproduct of a $k_\omega$-space and a discrete space.

On topological groups containing a Fr\'echet-Urysohn fan