Good Math/Bad Math

Friday, April 21, 2006

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.

8 Comments:

  • Your use of the word "function" implies that it must be one-to-one or one-to-many (since there is no such thing as a many-to-one function or many-to-many function). Am I right?

    By Blogger Thomas Winwood, at 3:11 PM  

  • thomas:

    Yes, but it never hurts to be redundant about these things. :-)

    By Blogger MarkCC, at 3:30 PM  

  • I think that the second example you gave, using the pentagon, would be better explained under the concept of a group acting on a set, and not as a group acting on itself.

    By Anonymous ParanoidMarvin, at 7:18 PM  

  • marvin:

    You're probably right; I was trying to avoid introducing yet another concept. Applying a group to a set is another idea on top of the basic concept of groups and what symmetry transformations are. The symmetry transformation of the pentagon is probably best seen as either the group applied to a set, or as a piece of a group permutation, rather than as a complete group permutation.

    By Blogger MarkCC, at 8:21 PM  

  • By the way, as another example of a group, and one that arises untrivially, I think a post on the fundamental group of topological spaces would be great. I've found that if you stay away from pathological spaces (i.e. talk about punctured discs) and from the technical details of homotopy equivalence of paths, this is a concept that can be explained easily to laymen, and provides an excellent example of use of groups.

    By Anonymous ParanoidMarvin, at 4:28 AM  

  • Nice writeup. I've got just one stupid question. Why is the homomorphism above defined as:

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

    instead of:

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

    ?

    Also, can you give an example of f above that would hold true for the two groups? The only one I could think of was f(x) = x * 0... Are there others that are homomorphisms for these groups and operations?

    By Anonymous D Kruz, at 9:38 PM  

  • d kruz:

    Remember that the "+" and "*" in the definition of homomorphism are not the arithmetic addition and multiplication operators. They're the group operations of two groups.

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

    Is a statement that the homomorphism preserves the group properties: if you use the group property of group A to compute "x+y", and then use the homorphism to map it to group B, then you get the same result as if you first mapped x and y to group B separately, and then used group B's operator.

    I think that that's likely a confusing enough issue that I should probably mention it in a top-level post; I'll do an example there as well.

    By Blogger MarkCC, at 10:00 PM  

  • Thanks, Mark! I'm following you, now.

    Informative and fun, as always.

    By Anonymous D Kruz, at 11:27 PM  

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