# Group Theory: How to find all possible images of $f$?

Let $H$ be a group and suppose that $f: D_{10} \rightarrow H$ is a homomorphism. How do I describe and justify all the possible images of $f$.

$D_{10} = ({1, \sigma, \sigma^2, \sigma^3, \sigma^4, \tau, \sigma\tau, \sigma^2\tau, \sigma^3\tau, \sigma^4\tau})$ where $\sigma = (12345)$ and $\tau = (13)(45)$. I've been stuck on this for a while. I was given a hint to use the 1st Isomorphism Theory but that fell through. Please help! I have found that normal subgroups of $D_{10}$ are $\left \{ e \right \}, \left \{ D_{10} \right \} and \left \langle \sigma \right \rangle$ but I've hit a wall.

Added: I've learnt that the First Isomorphism Theorem tells us that if $f: G \rightarrow H$ is a homomorphism then $G/Ker(f) \cong Im(f)$. Thus in the current example, as $f: D_{10} \rightarrow H$ has been defined as a homomorphism then $D_{10}/Ker(f) \cong Im(f)$. I've also been informed that the Kernels are the normal subgroups of $D_{10}$ which I found were $\left \{ e \right \}, \left \{ D_{10} \right \} and \left \langle \sigma \right \rangle$. So I need to find $D_{10}/\left \{ e \right \}, D_{10}/\left \{ D_{10} \right \}$ and $D_{10}/\left \langle \sigma \right \rangle$. But I'm stuck as to how.

Hint: Recall that the First Isomorphism Theorem tells you that if $f:G\to H$ is a homomorphism, then $G/\ker f\cong \text{Im } f$. I assume $H$ can be any arbitrary group, so we just need to think about what possible kernels could be. But remember that these are just the normal subgroups of $G$ (they are kernels of, for instance, the natural projections onto the induced quotient group). Find the normal subgroups of $G$ and mod out by them to find all possible images.
• Apologies, I've added the normal subgroups of $D_{10}$ that I've found. – Geometry Dec 4 '15 at 22:44
• You have all of them, so that's good! Now you can easily tell what the images are. For the case of $\{e\}$ and $\{D_{10}\}$, you should be able to find the trivial quotients. For $D_{10}/\langle \sigma\rangle$, what is the order of this quotient? What is the only group of that order? – J.G Dec 4 '15 at 22:49
• In this case the map is from $D_{10} \rightarrow H$, could you give me an example of an image? I think I'm having trouble getting over the fact that it's just mapping to $H$ with no other description. – Geometry Dec 4 '15 at 23:39
I think you must find all the normal subgroups of $D_{10}$ then use the fact that every normal subgroup is the kernel of a group homomorphism of $D_{10}$. i.e. $ker f = N$, where $N \unlhd D_{10}$. I hope that answer your question.