Homomorphism Construction Using Symmetry Groups

[Demon117]

If you have the dihedral group D4 and the symmetric group S8 how do you come up with a 1-to-1 group homomorphism from D4 to S8. I know what the multiplication table looks like. How can I use that to create the homomorphism?

Let R1,. . . ., R4 represent the rotation symmetry. Let u1, u2 represent the cross-sectional symmetries, and d1,d2 represents the diagonal symmetries.

A group homomorphism must take the form that for a,b in D4

(ab)phi = (a*phi)(b*phi)

But this is elusive notation. What does it mean and how can I proceed? An example to work off of similar to this would be great!

 

[micromass]

Let D8={1,a,a²,a³,b,ab,a²b,a³b}. Now try sending a to (1 2 3 4) and send b to (1 4)(2 3). If I'm not mistaken, that is a homomorphism...

 

[Outlined]

You can use the following proposition to check for 1-to-1:

a homomorphism f is an injection (= 1-to-1) <=> kernel f = 0