On large sequential groups

TL;DR

This paper constructs a special sequential group with unique properties regarding its subgroups and quotients, and explores the structure of $k_$ sequential groups, revealing conditions for metrizability and the presence of sequential fans.

Contribution

It provides a novel construction of a sequential group with specific subgroup and quotient properties, and advances understanding of $k_$ sequential groups' structure.

Findings

Constructed a sequential group with only closed, discrete countable subgroups.

Showed that $k_f$ sequential groups are either metrizable or contain a sequential fan.

Proved that dense proper subgroups of non-Fréchet $k_f$ groups are not sequential.

Abstract

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 Fr\'echet. We also show how to construct such a $G$ with several additional properties (such as make $G^2$ sequential, and arrange for every sequential subgroup of $G$ to be closed and contain 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 Fr\'echet $k_\omega$ sequential group is not sequential extending a similar observation of T.~Banakh about countable $k_\omega$ groups.