Please help me with these questions from abstract algebra Langrange theorem

Question

please help me with these questions from abstract algebra…. Langrange theorem…

Ans :−¿
Let f :G → H be a homomorphism of groups∧define the kernel of f by
ker ( f )={ g ∈G| f ( g ) =e }
Claim that ker ( f ) is subgroup for G
Subgroup test :−¿
Since G is group so…
Math