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I've been asked to find all group homomorphisms from $$\Bbb Z_n\to D_m,$$ where $n$ and $m$ are distinct natural numbers.

I now understand how to describe homomorphic groups using functions between two groups of numbers, but I'm a little confused how I would write a homomorphism from integers in $\Bbb Z$ mod $n$ to set permutations of symmetries on a regular shape in the dihedral group. Would I just represent the homomorphism as mapping, with arrows drawn from elements in $\Bbb Z_n$ to elements in $D_m$?

Thanks in advance for any help!

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    $\begingroup$ The image of a homomorphism $\mathbb Z_n\to G$ with any group $G$ is always cyclic of order $k$, where $k\vert n$, and any such cyclic subgroup of $G$ is also the image of a homomorphism from $\mathbb Z_n$. So you're essentially trying to find all cyclic subgroups of $D_m$ whose order divides $n$. Then find the homomorphisms from $\mathbb Z_n$ to those subgroups. $\endgroup$ Commented Jul 26, 2020 at 16:01
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    $\begingroup$ A homomorphism $f$ from $\mathbf Z/n\mathbf Z$ to a group $G$ is completely determined by $f(1)$, which must have order dividing $n$, so you need to know the orders of all elements in dihedral groups. $\endgroup$
    – KCd
    Commented Jul 26, 2020 at 16:02

3 Answers 3

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Every homomorphism is determined by where the generator of the cyclic group goes. There are $2m$ elements of the dihedral group. So you should consider $2m$ cases. In fact since $m$ elements are involutions and the rest forms a cyclic group of order $m$, you only should consider 2 cases since homomorphisms into a cyclic group you already know.

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  • $\begingroup$ Hi! Thanks for your response, but I'm not entirely sure what you mean by "you only should consider 2 cases since homomorphisms into a cyclic group you already know." Could you try expanding on this please? TIA! $\endgroup$ Commented Aug 1, 2020 at 2:26
  • $\begingroup$ I seem to have found 1 cyclic subgroup of D (with order "k" which satisfies k|n, where n is the order of Z), but I am thrown off a little by this "second case" that you bring up. $\endgroup$ Commented Aug 1, 2020 at 2:30
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  • $D_m$ is a semidirect product of $C_2$ with $C_m$.

  • A homomorphism from $C_n \to D_m$ is completely determined by its action on the generator of $C_n$.

  • The generator of $C_n$ has order $n$, so must be mapped to an element of order $d | n$.

The problem is now to find elements of $D_m$ with order $d | n$.

If $n$ is even you can map the generator to the 'swap' element of $D_n$.

You can map the generator to any of the 'rotate' elements of order $d | \gcd(m,n)$ which exist for every $d$.

Finally you can map the generator to an element of the form $s r^i$.

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Given $\Bbb Z_n$ is isomorphic to the group given by

$$\langle x\mid x^n\rangle$$

and $D_m$ is isomorphic to the group given by

$$\langle a,b\mid a^m, b^2, ba=a^{-1}b\rangle,$$

note that homomorphisms are determined by their actions on generators.

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  • $\begingroup$ The dihedral group should have order $2m$, not $2n$. Reread the question. $\endgroup$
    – KCd
    Commented Jul 26, 2020 at 16:03
  • $\begingroup$ Thank you, @KCd; I have corrected it. $\endgroup$
    – Shaun
    Commented Jul 26, 2020 at 16:04

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