Isomorphisms

Most every branch of mathematics concerns itself with its own special type of mathematical "objects," and there are a lot of them - groups, modules, topological spaces, manifolds, rings, fields, algebras, and vector spaces are just the ones that I can think of off the top of my head - and almost all of them define their own version of the word "isomorphism." In general, an isomorphism is a special type of function between two of the same "objects" that tells us when we can consider them eqivalent. As we will see, the existence of a group-isomorphism between two groups means that we can think of them as the same.

Before learning about isomorphisms, I should talk about bijections. I've used the word bijection before when talking about permutations, but in this post I'm going to need to actually show that things are both injective and surjective, so in order to make sure the meanings of those things are clear, I'm going to define them here.

Definition: Injective
Suppose φ:A→B. φ is called injective if different elements a,b∈A correspond to different elements, φ(a),φ(b)∈B. Symbolically, φ is injective if ∀a,b∈A, φ(a) = φ(b) implies that a = b or equivalently, φ is injective if ∀a,b∈A, a ≠ b implies that φ(a) ≠ φ(b). We call such functions injections.

Definition: Surjective
Suppose φ:A→B. φ is called surjective if the range of φ covers all of B. Symbolically, φ is surjective if ∀b∈B, ∃a∈A such that φ(a) = b. We call such functions surjections.

Definition: Bijective
A function is called bijective if it is both injective and surjective. Such functions are called bijections.

Now that we have the definition of a bijection, we're ready to define a group-isomorphism. Although the technical definition is that of a "group-isomorphism," when the context is known (that is when it is clear that we're talking about groups) I will leave out the word "group" and just refer to them as "isomorphisms."

Definition: Group-Isomorphism
If A and B are groups and φ:A→B is a function from A to B, then φ is an isomorphism if it is a bijection and ∀a,b∈A, φ(a∙b) = φ(a)∙φ(b). If there exists an isomorphism between two groups, A and B, then we say that A and B are isomorphic and write A≈B.

There are two things important in the definition of an isomorphism. First, it must be a bijection, which means that it is both injective and surjective, like always. The second, and more important part, is that it is what we call "operation preserving," which is the condition that ∀a,b∈A, φ(a∙b) = φ(a)∙φ(b). What this says is that if φ is an isomorphism between A and B, then when a,b∈A and φ(a),φ(b)∈B, then it doesn't matter whether you combine them in A or in B, you'll get the same thing on either side of φ. This idea of φ being operation preserving is shown pictorally, below. The dashed arrows represent the group operation and the solid arrows represent the mapping by φ.

In an effort to clarify all of this, here's an example.

Example:
Let A be the group of real numbers under addition and let B be the group of positive real numbers (not including zero) under multiplication. Let φ:A→B be defined by φ(x) = 2^x.

Step 1: φ is injective

Choose x,y∈A such that φ(x) = φ(y). Then we have 2^x = 2^y. Taking log₂ of each side gives log₂(2^x) = log₂(2^y) and by properties of logarithms we get that x = y. Therefore φ is injective.

Step 2: φ is surjective.

Choose y∈B. Let x = log₂(y). We note from the definition of logarithms that x∈A and then φ(x) = 2^log₂(y) = y. Therefor φ is surjective.

Step 3: φ is operation preserving.

Choose x,y∈A. Observe that φ(x+y) = 2^x+y = 2^x∙2^y = φ(x)∙φ(y). Therefore φ is operation preserving.

Steps 1 through 3 verify that φ is indeed an isomorphism and A≈B.

I don't imagine the usefulness of this concept of isomorphism is clear yet, but hopefully you can at least understand the definition. To sum things up a little bit, two groups are isomorphic if each element in one group corresponds to exactly one element in the other group, and combining two elements in the first group is the same as combining their corresponding elements in the second group. As we will see in the next post, there are a lot of very interesting properties that we get out of isomorphisms.

References

Previous Related Post: Recap - Subgroups

Text Reference: Gallain Chapter 6

Wikipedia: Group Isomorphism