On large sequential groups
We construct, using $\diamondsuit $, an example of a sequential group $G$ such that the only countable sequential subgroups of $G$ are closed and discrete, and the only quotients of $G$ that have a countable pseudocharacter are countable and Fréchet. We also show how to construct such a $G$ with several additional properties (such as $G^2$ being sequential, every sequential subgroup of $G$ being closed and containing a nonmetrizable compact subspace, etc.).
Several results about $k_\omega $ sequential groups are proved. In particular, we show that each such group is either locally compact and metrizable or contains a closed copy of the sequential fan. It is also proved that a dense proper subgroup of a non-Fréchet $k_\omega $ sequential group is not sequential, which extends a similar observation of T. Banakh about countable $k_\omega $ groups.