8. Maps Between Groups

Sometimes, two groups can be considered essentially the same even if they look different.

Example

The subgroup $\{Id, r, r^2\}$ of $D_3$ is essentially the same as the cyclic group $\dfrac{\Z}{3}$

We can define a map $\phi$ between their elements

$$\phi : \begin{cases} Id \mapsto 0 \pmod{3} \\ r \mapsto 1 \pmod{3} \\ r^2 \mapsto 2 \pmod{3} \end{cases}$$

Respects group operations

$$\phi(r \circ r^2) = \phi(Id) = 0$$ $$1 + 2 \mod 3 = 0$$ $$\phi(r) + \phi(r^2) = 0$$

it can help to read through each operation slowly to grasp the idea

 

Homomorphisms

Let $G$ and $H$ be groups with operations $\cdot_G$ and $\cdot_H$

Let $\phi: G \to H$

A function is a homomorphism if

$$\phi(x \cdot_G y) = \phi(x) \cdot_H \phi(y) \quad \text{for all } x, y \in G$$

Bijection

A function that pairs each element of one set with exactly one element of another set, with no elements left unmatched in either set.

Isomorphsim

If $\phi$ is also a bijection, it is called an isomorphsim, and $G$ and $H$ are said to be isomorphic

Isomorphic groups have the same structure, even if their elements look different