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I'm currently studying up on homomorphisms and I stumbled upon a problem that I've had some trouble coming up with a proof for. In the problem we have $G$ being a finite abelian goup and $n \in \mathbb{N}$. We then define a map $\phi:G \to G$ by $\phi(g) = ng$, for all $g\in G$. The problems asks to show that $ker(\phi) \cong G/im(\phi)$.

The case where $n$ does not divide $|G|$ is straight forward since in that case we know the only element that will map to the identity is the identity since the order of every element of $G$ must divide $|G|$. Thus our kernel is trivial and hence the map is one-to-one which implies the map is also surjective and thus $im(\phi) = G$. Thus in this case it is easy to see that $ker(\phi) \cong G/im(\phi)$.

I am having a problem when $n$ divides $|G|$. The kernel should be all elements whose order is equal to $n$ and the identity but that is where I am getting stuck. I don't know how to proceed. I would be appreciative of any suggestions.

Thanks.

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  • $\begingroup$ $n$ not dividing $|G|$ does not mean that $ord(g)$ does not divide $n$. We might still have a non-trivial kernel. For example, $G=\mathbb{Z}_6$, $n=4$. In this case $3\in \ker(\phi)$ $\endgroup$ – Dan Jan 4 '18 at 18:06
  • $\begingroup$ Oh I see that now. I should of thought about a case like that. $\endgroup$ – user1234 Jan 4 '18 at 18:16
  • $\begingroup$ Is the group operation, addition? $\endgroup$ – Abishanka Saha Jan 9 '18 at 4:29
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Hint: put $d= \frac{|G|}{\gcd (|G|, n)}$. What can you say about $x\mapsto dx$ ?

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  • $\begingroup$ I'm not exactly sure. I had a couple ideas when I tried some different cases of $G = \mathbb{Z}_k$ and $n$ but when I considered $\mathbb{Z}_2 \times \mathbb{Z}_2$ most of those ideas didn't work. $\endgroup$ – user1234 Jan 5 '18 at 16:48
  • $\begingroup$ Do you see that it's a morphism ? Can you determine its image and its kernel ? $\endgroup$ – Max Jan 5 '18 at 16:51
  • $\begingroup$ In my examples the kernel appears to be elements whose order divides $gcd(|G|,n)$. The image of the map I am not as certain about. $\endgroup$ – user1234 Jan 5 '18 at 16:58
  • $\begingroup$ Are you certain of that ? Imagine $n$ and $|G|$ are coprime, so that $d= |G|$ : do you think the kernel is $\{0\}$ ? In fact I'm surprised you'd even find an example where this is the case $\endgroup$ – Max Jan 5 '18 at 17:30
  • $\begingroup$ I see what you mean. If $d = |G|$ then our kernel would be all of $G$. $\endgroup$ – user1234 Jan 5 '18 at 17:35

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