Definition of homomorphism of groups
Let (G,*) and (G_{1},o) be two groups and f be a function from G to G_{1}. Then f is called a homomorphism of G into G_{1} if for all a,b∈G
f(a*b) = f(a) o f(b)
To do the above problem you have to know the following theorem
If f is a homomorphism from a group G into a group G_{1} and e,e_{1} are the identity elements of G and G_{1} respectively ,then
(i) f(e) = e_{1}
(ii) f(a^{-1}) = f(a)^{-1 }for all a ∈ G
Now, come to the given problem
Suppose f is a monomorphism. Then f is injective.
Let a∈ ker f and also assume e be the identity element of G and e_{1} be the identity element of G_{1}.
Hence, f(a) = e_{1} ( From the definition of Ker f)
Again f(e) = e_{1 } ( From the theorem written above)
Since f is injective we have a=e.
Hence ,ker f = {e}.
Now, to prove the converse
Conversely, suppose ker f = {e}.
Let x,y ∈ G such that f(x) = f(y)
Then f(x)f(y)^{-1} = e_{1}
=> f(x)f(y^{-1}) = e_{1} (From the theorem written above)
=> f(xy^{-1}) = e_{1} (From the definition of homomorphism of groups).
=> xy^{-1} ∈ ker f = {e} and so xy^{-1} = e.
Hence, x=y.
Therefore f is injective and so f is a monomorphism.
Hence the problem
A homomorphism f:G →G_{1} of groups is a monomorphism iff Ker f = {e}.