Definition of homomorphism of groups
Let (G,*) and (G1,o) be two groups and f be a function from G to G1. Then f is called a homomorphism of G into G1 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 G1 and e,e1 are the identity elements of G and G1 respectively ,then
(i) f(e) = e1
(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 e1 be the identity element of G1.
Hence, f(a) = e1 ( From the definition of Ker f)
Again f(e) = e1 ( 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 = e1
=> f(x)f(y-1) = e1 (From the theorem written above)
=> f(xy-1) = e1 (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 →G1 of groups is a monomorphism iff Ker f = {e}.