Group generated by a nonzero translation

[zcdfhn]

Let G₁ be the group generated by a nonzero translation and G₂ be the group generated by a glide reflection. Show that G₁ and G₂ are isomorphic.

Here is how I started:

G₁ = <T_b> where b$$\in$$ C and T_b(z) = z+b

G₂ = <M_L $$\circ$$ T_c> where c and L are parallel to each other.

Let's define a function $$\Phi$$: G₁ $$\rightarrow$$ G₂

Then if $$\Phi$$ is a homomorphism and a bijection, it is an isomorphism.

But here lies my problem, I do not know what to make $$\Phi$$ equal to. Maybe this isn't the right way of approaching this problem.

Thanks in advance.

 

[e(ho0n3]

How can you perform a translation using a glide reflection? How can you perform a glide reflection with a translation?

 

[zcdfhn]

A glide reflection composed with itself is a translation. And you get a glide reflection by composing a reflection with a translation.

 

[e(ho0n3]

So informally, translation = 2 * glide reflection and glide reflection = reflection + translation. Since you need a reflection to get a glide translation, I don't see any way of getting an isomorphism between the two groups.