kernel of subgroup homomorphism

Question

Let $f:A \longrightarrow B$ be a group homomorphism, and note $C$ a subgroup of $A$ and $D$ a subgroup of $B$. Can we find a link between the kernel of $f$ and the kernel of the group homomorphism $g: C \longrightarrow D$, assuming $\forall c \in C, g(c) = f(c)$? Like $ker(g)= ker(f) \cap C$. I don't know books containing some information about the kernel of homomorphism of subgroups

Answer 1

(I include in this the additional assumptions mentioned in the comments.)

Prove both directions of containment: one direction is trivial since if $x\in \ker(g)$, then $g(x) = 0 \Rightarrow f(x) = 0$ and by definition $x\in C$.

So let's look at $\ker(f) \cap C \subset \ker(g)$. For an element $x$ in the LHS you need to show $g(x) = 0$. Well, by hypothesis, $x\in C \Rightarrow g(x) = f(x)$, and also by hypothesis, $x\in \ker(f)$. Thus, $g(x) = f(x) = 0$ and so indeed $x\in \ker(g)$.

Thus, $\ker(g) = \ker(f) \cap C$.