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

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!

• 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. Commented Jul 26, 2020 at 16:01
• 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.
– KCd
Commented Jul 26, 2020 at 16:02

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.

• 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! Commented Aug 1, 2020 at 2:26
• 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. Commented Aug 1, 2020 at 2:30
• $$D_m$$ 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 $$s r^i$$.

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.

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