proof of properties of Hopfian and co-Hopfian groups
Proof. “” Assume that is a surjective homomorpism such that is not an automorphism, which means that is nontrivial. Then (due to the First Isomorphism Theorem) is isomorphic to . Contradiction, since is Hopfian.
“” Assume that is not Hopfian. Then there exists nontrivial normal subgroup of and an isomorphism . Let be the quotient homomorphism. Then obviously is a surjective homomorphism, but is nontrivial, therefore is not an automorphism. Contradiction.
Proposition. A group is co-Hopfian if and only if every injective homomorphism is an automorphism.
Proof. “” Assume that is an injective homomorphism which is not an automorphism. Therefore is a proper subgroup of , therefore (since and due to the First Isomorphism Theorem) is isomorphic to its proper subgroup, namely . Contradiction, since is co-Hopfian.
“” Assume that is not co-Hopfian. Then there exists a proper subgroup of and an isomorphism . Let be an inclusion homomorphism. Then is an injective homomorphism which is not onto (because is not). Contradiction.
|Title||proof of properties of Hopfian and co-Hopfian groups|
|Date of creation||2013-03-22 18:31:17|
|Last modified on||2013-03-22 18:31:17|
|Last modified by||joking (16130)|