Let be a free group (not necessarily finitely generated). I am trying to prove that has infinite index, and I have, so far, shown that this is equivalent to proving that contains no torsion. Which is obvious.
Let be a free group (not necessarily finitely generated). I am trying to prove that has infinite index, and I have, so far, shown that this is equivalent to proving that contains no torsion. Which is obvious.
Except...I can't seem to prove it...
is simply a free abelian group. A free abelian group is the internal direct sum of a family of infiinite cyclic subgroups, so it is torsion-free.