On Isomorphic Groups and Homeomorphic Spaces

Let $C(X,G)$ denote the group of continuous functions from a topological space $X$ into a topological group $G$ with the pointwise multiplication. Some classes of SQ-pairs and properties of the corresponding topological group $C(X,G)$ with the compact-open topology are investigated. We also show that the existence of a group isomorphism between groups $C(X,G)$ and $C(Y,G)$ implies the existence of a homeomorphism between $X$ and $Y$, if $(X,G)$ and $(Y,G)$ are SQ-pairs.