Image and Kernel of composition of two homomorphisms

Question

I have just showed that the composition of $a * b$ of two homomorphisms $a,b$ is a homomorphism. However, what can I say about the image and kernel of $a*b$, in terms of $a$ and $b$? Is there something special about this image and kernel now that there is a composition of two homomorphisms?

Answer 1

$\newcommand{\Set}[1]{\left\{ #1 \right\}}$ First of all, I would suggest you use $a \circ b$ as the standard symbol for composition.

Then note that if $b : G \to H$ and $a : H \to K$ are morphisms of multiplicatively written groups, say, you have $$ \ker(a \circ b) = \Set{g \in G : a(b(g)) = 1} = \Set{g \in G : b(g) \in \ker(a)} = b^{-1}(\ker(a)). $$ Here $b^{-1}$ is not the inverse of $b$, which need not exist, but it is the map that takes a subset $L$ of $H$ to the subset $$ b^{-1}(L) = \Set{ g \in G : b(g) \in L} $$ of $G$.

As to the image, you have simply $$ a \circ b(G) = a(b(G)), $$ so the image of $a \circ b$ is the image under $a$ of the image $b(G)$ of $b$. This is true at the level of sets.