Isomorphisms of $GF(2)$

Question

The additive group of $GF(2)$ is isomorphic to $\mathbb{Z}2/\mathbb{Z}$ under addition with the "carryless" addition taken modulo 2.

An appropriate relabelling of the elements ($0 \rightarrow 1$ and $1 \rightarrow -1$) maps the elements of $\mathbb{Z}2/\mathbb{Z}$ onto $\mathbb{C}$ where addition corresponds to complex multiplication. This is an isomorphism to $\mathbb{C}$.

Another isomorphism maps elements of $GF(2)$ to elements of $\mathbb{Z}/3\mathbb{Z}^{\times}$, where addition becomes multiplication modulo 3 ($\mathbb{Z}/3\mathbb{Z}^{\times}$ having $\phi(3)=2$ elements).

• Are there any other, perhaps less-obvious, isomorphisms that I have not included?
• Are there any fully isomorphic mappings that facilitate both addition and multiplication in another structure?
• Why do isomprhisms like this exist?

I'm looking for reading material, so any mention of book titles would be very helpful.

Answer 1

Basically by construction, for any group $\mathbf{G}$, there is a natural bijective correspondence between:

• Group homomorphisms $\mathbf{Z} / 2 \mathbf{Z} \to \mathbf{G}$
• $2$-torsion elements of $\mathbf{G}$

Furthermore, the homomorphism is monic if and only if the $2$-torsion element is not the identity.

So because lots of groups have 2-torsion elements, you find lots of homomorphisms from $\mathbf{Z} / 2 \mathbf{Z}$.