An asymmetric coloring of a graph is its vertices that not preserved by any non-identity automorphism the graph. The motion minimal degree group, i.e., minimum number elements are moved (not fixed) automorphism. We confirm Tom Tucker's “Infinite Motion Conjecture” connected locally finite graphs with infinite admit an 2-coloring. infer this from more general result inverse limit sequence permut...

Akbarov’s theory of holomorphic reflexivity for topological Hopf algebras has been developed in two directions, namely, by the complication definitions when expanding scope and their simplification restricting. In framework latter approach, we establish associated with locally finite countable groups second-countable profinite groups. Abelian case, is described terms close to classical ones.