I am completing this question as part of my study for my upcoming mid-term:
Find an isomorphism between (C, +) and (F, o). Justify your answer
- C is the set of complex numbers
- ∀a ∈ C, the map fa : C → C given by f(z) = z+a namely the translation by a
- Collection of maps F is defined:
The question I am trying to answer above is part c) of a three-part question. Parts a and b asked to show (C, +) and (F, o) were groups and I did this by showing they followed the axioms of closure, associativity, identity element, and inverse to which I concluded both were in fact groups.
I am unsure how to find an isomorphism between the two groups any help would be appreciated, thanks in advance!
Consider $\phi: \mathbb{C} \to \mathcal{F}_\mathbb{C}$ defined trivially: $\phi(a) = f_a$. You should convince yourself that such a function is well-defined.
We must now show that $\phi$ an isomorphism.
First, we show that $\phi$ is a bijection. Given $\phi(a_1) = \phi(a_2)$, it means that the function $f_{a_1} = z + a_1$ is equal to the function $f_{a_2} = z + a_2$. Of course, this implies $a_1 = a_2$, so $\phi$ is injective. Next, take an arbitrary element of $\mathcal{F}_\mathbb{C}$, $f_b = z + b \implies b \in \mathbb{C}$. Note that $\phi(b)$ is precisely $f_b$, and since $f_b$ was arbitrary, $\phi$ must be surjective.
Next, we show that $\phi$ is a homomorphism. For this, we need to show that $\phi (a_1 + a_2) = \phi (a_1) \circ \phi (a_2)$ for arbitrary $a_1, a_2 \in \mathbb{C}$. Observe that $\phi (a_1 + a_2) = f_{a_1 + a_2} = z + a_1 + a_2$, and $\phi (a_1) \circ \phi (a_2) = f_{a_1} \circ f_{a_2} = (z + a_2) + a_1 = z + a_1 + a_2$. They are equal, so $\phi$ preserves the group structures and is a homomorphism.