The groups of the semigroup of compact subsets of a topological group (Q802762)

The compact subsets of a topological group G form a topological semigroup S(G) under set product and Vietoris topology. The author shows that the subgroups of S(G) are precisely the topological groups of the form H/E, where \(E\trianglelefteq H<G\), E is compact, and H/E has the quotient topology. Conditions are given for these subgroups to be open or closed. The author also points out that the Schützenberger group of a D-class of S(G) is topologically isomorphic to a section of G by a compact subgroup and the Schützenberger groups of S(G) are topological groups.