  A topological semigroup is monothetic provided it contains a dense cyclic
subsemigroup. The Koch problem asks whether every locally compact monothetic
monoid is compact. This problem was opened for more than sixty years, till in
2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive
answer for Koch's problem for some special classes of topological monoids.
Namely, we show that a locally compact monothetic topological monoid is a
compact topological group if and only if $S$ is a submonoid of a
quasitopological group if and only if $S$ has open shifts if and only if $S$ is
non-viscous in the sense of Averbukh. The last condition means that any
neighborhood $U$ of the identity $1$ of $S$ and for any element $a\in S$ there
exists a neighborhood $V$ of $a$ such that any element $x\in S$ with $(xV\cup
Vx)\cap V\ne\emptyset$ belongs to the neighborhood $U$ of 1.

Group Theory