  A.V.Arkhangel'skii asked in 1981 if the variety $\mathfrak V$ of topological
groups generated by free topological groups on metrizable spaces coincides with
the class of all topological groups. We show that if there exists a real-valued
measurable cardinal then the variety $\mathfrak V$ is a proper subclass of the
class of all topological groups. A topological group $G$ is called
$g$-sequential if for any topological group $H$ any sequentially continuous
homomorphism $G\to H$ is continuous. We introduce the concept of a
$g$-sequential cardinal and prove that a locally compact group is
$g$-sequential if and only if its local weight is not a $g$-sequential
cardinal. The product of a family of non-trivial $g$-sequential topological
groups is $g$-sequential if and only if the cardinal of this family is not
$g$-sequential. Suppose $G$ is either the unitary group of a Hilbert space or
the group of all self-homeomorphisms of a Tikhonov cube. Then $G$ is
$g$-sequential if and only if its weight is not a $g$-sequential cardinal.
Every compact group of Ulam-measurable cardinality admits a strictly finer
countably compact group topology.
