Describing homomorphisms from $\Bbb Z_n$ to $D_m$.

Question

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!

Answer 1

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.

Answer 2

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.

Answer 3

• $D_m$ 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 $s r^i$.