I am trying to show that F2 (the free group on 2 elements) has an infinite index, free subgroup of rank 2.
As it has to be a free subgroup of rank 2 I'm guessing I need to find 2 elements in F2 that generate a free subgroup together (satisfying no relations between them), and then also add in the constraint that the group they generate has infinite index in F2.
However, I'm really struggling to get my head around what it means for a group to have 'infinite index' and how, even if I found such a pair of elements somehow, how I could prove that they had infinite index in F2.
Also, this question occurs before material about topological methods such as covering spaces, so I would like to answer this without such tools, so I guess purely algebraically.
Any hints, suggestions or indeed solutions would be much appreciated.