kilmo: FYI, this characterisation of the Cantor group is awkward compared to the following characterisation. Take all infinite binary sequences. This is an additive group with the xor operation. Define the topology as the smallest one such that the set of all sequences which have 0 in the i'th place is both closed and open.
With this chracterisation you don't get an embedding the Cantor group on the line, but you do get an easy way to show all kinds of nice things.
Exercise: can you give this set another continuous group structure such that each element has order 0?
Hint: Can you give the set of all binary sequences of length n a group structure such that there are elements of order 2^n?