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Homomorphisms and Surjections

Let f:G->H be a group homomorphism.
Prove or disprove the following statement.

1.Let a be an element of G. If f(a) is of finite order, then a is also of finite order.

2.Let f be a surjection. Then f is an isomorphism iff the order of the element f(a) is equal to the order of the element a , for all a belong to G.

First, I want to know if f is the trivial homorphism, then both will fail, right?

Second, if f is non-trivial homomorphism. Both will hold, right?

Finally, Please help me to prove both or give me counterexamples without considering the trivial homomorphism.

Solution Preview

1. False.
No matter f is a trivial or non-trivial group homomorphism, the statement is false. Here is a counter example. Let G=Z be the additive group of integers, H=Z2={0,1} is the additive group of Z mod 2. f:G->H is defined as f(a)=a mod 2 for all a in G. Then every nonzero element a in G has infinite order, but f(a) in H has order 0 or 1.

2. True.
No ...

Solution Summary

Homomorphisms and surjections are investigated. The solution is detailed and well presented.