Let be a group homomorphism and let be a subgroup. Prove that (the -preimage of ) is a subgroup of . If , prove that . Deduce that .
- Let . Then , and since is a subgroup, by the subgroup criterion. So , and by the subgroup criterion we have .
- Suppose is normal in . Now let and ; we have , since is normal. Thus , hence is normal in .
- is trivially normal for all , and by definition . So is normal in .