Showing that dihedral 4 is isomorphic to subgroup of permutations

[Mr Davis 97]

Homework Statement

D4 acts on the vertices of the square. Labeling them counterclockwise
starting from the top left as 1, 2, 3, 4, find the corresponding homomorphism
to S4.

Homework Equations

The Attempt at a Solution

I am not completely sure what the question is asking. It's pretty clear to see which elements of D4 would correspond to which permutations of S4, so am I being asked to just list which elements of D4 correspond to which permutations of S4? How would I prove the the resulting map is a homomorphism? It seems like it would be tedious to check all values to see if the homomorphism property is always satisfied.

 

[fresh_42]

Homework Statement

D4 acts on the vertices of the square. Labeling them counterclockwise
starting from the top left as 1, 2, 3, 4, find the corresponding homomorphism
to S4.

Homework Equations

The Attempt at a Solution

I am not completely sure what the question is asking. It's pretty clear to see which elements of D4 would correspond to which permutations of S4, so am I being asked to just list which elements of D4 correspond to which permutations of S4?

That's how I see it. In the end, it's all about how you define $D_4$. E.g. it can be defined as the group generated by two elements $r\, , \,s$ with the relations $r^2=s^2=(rs)^4=1$ where it's not immediately obvious how the representation by a monomorphism $\tau$ to $\mathcal{Sym}(4)$ works.

How would I prove the the resulting map is a homomorphism? It seems like it would be tedious to check all values to see if the homomorphism property is always satisfied.

Yes, that's true. Of course you could simply look it up on Wikipedia, or you try to find an argument, why the structure is preserved, i.e. why $\tau (a\cdot b) = \tau (a) \cdot \tau(b)$ holds and why $\tau$ is injective.
The elements of $D_4$ are rotations by $90°$ and reflections along both middle axis of a square. So you have to find an argument, why two such operations in succession $a\cdot b$ map (via $\tau$) to the same corresponding succession of permutations $\tau (a) \cdot \tau(b)\,.$

 

[Mr Davis 97]

Could the idea of group action help me out here?

 

[fresh_42]

Could the idea of group action help me out here?

Yes. Just make sure that it doesn't become a terminological overkill here. You don't need the entire apparatus of group operations.

Only ask what the elements in these groups are and how multiplication is defined on them. Is there any difference between the two sides of $\tau : D_4 \hookrightarrow \mathcal{Sym}(4)$ ? Or what is $\tau^{-1}(\tau (a) \circ \tau (b))$ ? The homomorphy lies already in the concept itself, if you define $D_4$ as a group of transformations instead of as a group with generators and relations. That's why I asked about the definition of $D_4$. This determines the amount of work that has to be done. If you meant this by group action, then the answer to your question is yes.