# Complement of Normal subgroups and free groups

Does every normal subgroup has complement in free groups? What about free abelian groups i.e. Is free abelian gorup complemented group?

Definition: If there exist a subgroup K such that HK = G and H ∩ K = {e}, then K is called complement of H. Here e is identity of G

• What do you mean by the complement of a normal subgroup? In which sense? – Malkoun Jun 13 '16 at 8:45
• ok. What is the complement of $2\mathbb{Z}$ in $\mathbb{Z}$? – Malkoun Jun 13 '16 at 9:22
• @Malkoun okay I got it. complement of 2Z doesn't exist because any other subgroup intersect it. – Sushil Jun 13 '16 at 9:37
• I am glad I could help! – Malkoun Jun 13 '16 at 9:39