If $G$ is an infinite group, is there necessarily an unbounded left-invariant metric on $G$? 



Unbounded metrics on groups
The answer is no. See Thm 1.2 of <http://homepages.math.uic.edu/~rosendal/PapersWebsite/Property(OB)10.pdf>. There is a property discussed in the intro of this paper which is equivalent to all left invariant metrics are bounded. It is known that certain large permutation groups have this property. 


