Nessary and sufficient condition for homomorphism to be isomorphism.

[AAQIB IQBAL]

The necessary and sufficient condition for homomorphisim f of a group G into a group G' with kernel K to be isomorphism of G into G' is that k={e}
... THOUGH I AM ABLE TO PROVE THAT f IS ONE-ONE AND f IS HOMOMORPHISM (in converse part) BUT CAN'T GET ANY IDEA TO PROVE THAT f IS ONTO.
PLEASE HELP ME IN THIS REGARD

 

[micromass]

That Ker(f)={e} is a necessary and sufficient condition for f to be injective.
You won't be able to prove that f is an isomorphism, because it is false in general.