Group Isomorphism: Defining Symmetry Transformations

Yesterday, I explained the idea of multiple kinds of symmetry intuitively, by describing symmetry as immunity to transformation. Today I'm going to try to explain what, in group theory transformation means, and how symmetry is defined in terms of transformation.

What it comes down to is a concept called group isomorphism.

Group Homomorphisms and Isomorphisms

Before I can describe a group isomorphism, I need to explain a group homomorphism. A homomorphism between two groups A and B is a function that maps members of A to members of B in a way that preserves the group property.

So - suppose we have group A, and we call its operation "+"; and we have a group B, and we call its operation "*". A function f from A to B (written f : A -> B) is a homomorphism if and only if:

all x,y in A : f(x+y) = f(x)*f(y)

So a homomorphism is a mapping in one direction from A to B. It guarantees that for every element of A, we can map it onto an element of B; and that that mapping is guaranteed to preserve the group property on the mapped elements.

What do I mean by the group property is preserved? Mostly that the statement up above is true: all x,y in A: f(x+y) =f(x)*f(y). But that implies a couple of important things:

• (1) f maps the identity element of A onto the identity element of B: f(1_A) = 1_B.
  
• (2) f's mapping preserves inverses: f(x^-1) = f(x)^-1.
  

A homomorphism does not have to be a total function onto B - meaning that non every member of B has an element of A mapped onto it. It does not have to be one-to-one, meaning that multiple elements of A could be mapped onto a single element of B, so long as the group property was preserved.

A group isomorphism is a group homomorphism f, where f is a bijection. That means that f maps each element of A onto exactly one element of B; and every element of B is mapped to by exactly one element of A. (Another way of saying that is f : A -> B is total, one-to-one, and onto.)

If f : A -> B is a group isomorphism, then it describes a symmetric transformation from group A to group B. If we take a set of values from A; and we map each of those values to B using f, we get as a result a set which is equivalent or indistinguishable from the original set.

Permutation Symmetry

One particularly interesting kind of symmetry is called permutation symmetry. Permutation symmetry is a kind of symmetry created by a group isomorphism from a group to itself. Reflection symmetry between the real numbers is a permutation symmetry: it is a total, one to one, onto map from all real values (omitting zero) to all real values (omitting zero) which preserves the group property of multiplication.

The rotational symmetry of a pentagon is also a permutation symmetry; if we label the five vertices of the pentagon as A, B, C, D, and E, then the function: { (A->B), (B->C), (C->D), (D->E), (E->A) } is an isomorphism; when you apply it, you rotate the pentagon (or the labeling of the pentagon), but you end up with a pentagon that is indistinguishable from the one you started with.