TITLE: Automorphism group of a topological space
QUESTION [8 upvotes]: Let $G$ be any group. Is there a topological space $(X,\tau)$ such that the automorphism group $\textrm{Aut}(X,\tau)$ is isomorphic to $G$?

REPLY [3 votes]: As pointed out in the comments, this has been answered by Tony Huynh on MathOverflow. In

de Groot, J. ($1959$), Groups represented by homeomorphism groups, Mathematische Annalen $138$

the author shows that:

"for every group $G$ one can find a complete, connected, locally connected metric space $M$ of any positive dimension such that $G \cong A(M)$"

where $A(M)$ denotes the autohomeomorphism group of $M$.