TITLE: Can you give me an example of topological group which is not a Lie group.
QUESTION [17 upvotes]: I know the definitions of Lie group and topological group are different. Can you give me an example of topological group which is not a Lie group.

REPLY [10 votes]: The $p$-adic numbers are a topological group, but not a Lie group. 
There are many profinite groups which are not Lie groups, for example the profinite group completion of a knot group.