Group Isomorphism
In abstract algebra, an isomorphism between two groups (G,·) and (H,*) is a one-to-one correspondence (bijection) between them. This means that for every element a and b in G, the following holds: $$ f(a·b) = f(a)*f(b) $$ where f(a) and f(b) are elements of (H,*).
Simply put, an isomorphism is a two-way homomorphism between two groups:
- a homomorphism from the first group to the second by the function $$ f:G \rightarrow H $$
- a homomorphism from the second group back to the first through the inverse function $$ f^{-1}:H \rightarrow G $$
A Practical Example
Take the exponential function:
$$ f(x) = e^x $$
Consider the additive group of real numbers
$$ (R,+) $$
and the multiplicative group of positive real numbers
$$ (R^+,·) $$
These two groups, (R,+) and (R+,·), are isomorphic through the function f(x)=ex because the following identity is true:
$$ \forall \ a,b \in R \Longrightarrow e^{a+b}=e^a \cdot e^b $$
The reverse is also true:
$$ \forall \ a,b \in R^+ \Longrightarrow \ln(a \cdot b)= \ln(a) + \log(b) $$
Here, the logarithm function (ln) is the inverse of the exponential function.
And so on.
