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