On two methods of constructing compactifications of topological groups

The classification of (proper) compactifications of topological groups with respect to the possibility of extensions of algebraic operations is presented. Ellis'method of construction compactifications of topological groups allows one to obtain all right topological semigroup compactifications on which the multiplication on the left continuously extends. Presentation of group elements as graphs of maps in the hyperspace with Vietoris topology allows one to obtain compactifications on which the involution and the multiplication on the left extend.