Describe subgroups of the free group \(F_2=\langle a,b\rangle\). Give examples having ranks \(3\), \(4\), and countably infinite rank. For rank \(4\), give both normal and nonnormal examples.
Solution
By the Nielsen–Schreier theorem, every subgroup of a free group is free. If \(H\le F_r\) has finite index \(d\), the Schreier index formula gives \[ \operatorname{rank}(H)=1+d(r-1). \] For \(F_2\), this becomes \[ \operatorname{rank}(H)=d+1. \]
1. Rank \(3\):Let\[ H_2=\ker(F_2\twoheadrightarrow C_2),\qquad a\mapsto 1,\quad b\mapsto0. \]Then \([F_2:H_2]=2\), so \(H_2\) is free of rank \(3\). It is normal because it is a kernel.
2. Rank \(4\), normal:Let\[ H_3=\ker(F_2\twoheadrightarrow C_3),\qquad a\mapsto1,\quad b\mapsto0. \]Then \([F_2:H_3]=3\), so \(H_3\) is free of rank \(4\), and it is normal.
3. Rank \(4\), nonnormal:Map \(F_2\) onto \(S_3\) by\[ a\mapsto(12),\qquad b\mapsto(123). \]Let \(K\) be the inverse image of \(\langle(12)\rangle\). Since that subgroup has index \(3\) in \(S_3\), \(K\) has index \(3\) in \(F_2\), hence rank \(4\). But \(\langle(12)\rangle\) is not normal in \(S_3\), so \(K\) is not normal in \(F_2\).
4. Countably infinite rank:The commutator subgroup\[ [F_2,F_2]=\ker(F_2\twoheadrightarrow\ZZ^2) \]is normal and has infinite index. It is a nontrivial subgroup of the countable group \(F_2\), hence is at most countably generated; by Nielsen–Schreier it is free. It cannot have finite rank: a nontrivial finitely generated normal subgroup of a finitely generated free group has finite index. Therefore \([F_2,F_2]\) is free of countably infinite rank.