Properties of Isomorphisms

Last time we learned the definition of an isomorphism. Today I intend on explaining the usefulness of this concept of isomorphic groups. The first thing I'm going to do is list a bunch of properties of isomorphisms without proof. You don't need to remember them, but I'd like to use them to help convince you of the vast consequences of this idea of isomorphic groups.

Theorem: Properties of Isomorphisms
Suppose that A and B are groups and φ is an isomorphism between them. Then

• If A and B are finite, then |A| = |B|.
• If e_A∈A is the identity of A and e_B∈B is the identity of B, then φ(e_A) = e_B.
• ∀n∈Z and ∀a∈A, φ(aⁿ) = [φ(a)]ⁿ.
• ∀a,b∈A, a∙b = b∙a if and only if φ(a)∙φ(b) = φ(b)∙φ(a).
• A = <a> if and only if B = <φ(a)>.
• ∀a∈A, |a| = |φ(a)|.
• ∀k∈Z and ∀a∈A, |{x∈A : x^k = a}| = |{x∈B : x^k = φ(a)}|. In other words, the number of solutions to the equation x^k = a is the same as the number of solutions to the equation x^k = φ(a).
• If A and B are finite, then then they have exactly the same number of elements of every order.
• The function φ^-1:B→A is an isomorphism from B to A.
• A is abelian if and only if B is abelian.
• A is cyclic if and only if B is cyclic.
• If H is a subgroup of A, then the set φ(A) = {φ(h) : h∈H} is a subgroup of B.

Normally, when I give a theorem, I then try and explain it, but I'm not really going to do that this time. These 12 facts are more interesting than useful, though they are sometimes useful in proving that groups are not isomorphic. For example, one might conjecture that Z(6) ≈ S₃ because |Z(6)| = 6 = |S₃|, but it would be very easy to calculate the orders of every elements of both groups, and show that these groups fail the 8^th property above.

However, the reason that I listed all of these properties of isomorphisms is to make it evident that isomorphic groups have many properties in common. In fact, isomorphic groups have all of their group-theoretic properties in common. And more than that, the definition of a group-isomorphism was precisely formulated so that isomorphic groups would share every group-theoretic property. What this means is that suppose I have two groups that are isomorphic, but I cannot actually see the individual elements - rather I can only see their interactions. Then I could not tell the difference between them. Because of this, mathematicians tend to think of isomorphic groups as "equal" or as "the same." This is the real power of the idea of isomorphism because it greatly reduces the number of groups that we need to study. For example, it turns out that S₃ ≈ D₃ (this is a very special case - it is not even close to true that S_n ≈ D_n for n≠3). That means that there is no real point in studying both groups in detail because whatever we know about one transfers automatically to the other. As an even broader example, suppose that G is cyclc and |G|=n. Then by a very simple argument we can show that G ≈ Z(n). The cyclic groups, Z(n) are very well understood, so as soon as we know that a group is cyclic, we know everything there is to know about that particular group. As you can see, isomorphisms very much reduce the amount of work that we have to do as group theorists.

Before we leave, there's a couple of other things we should discuss about isomorphisms. Primarily, there are a couple of special sorts of isomorphisms that have their own names.

Definition: Automorphism
An isomorphism from a group G onto itself is called an automorphism on G.

It should be clear that a group is isomorphic to itself. First of all, it makes sense - that is, we defined the notion of isomorphism to be such that isomorphic groups are the same, and clearly a group should be the same as itself. If you'd like to be rigorous, though, if we take a group, G, and define φ:G→G by φ(g)=g ∀g∈G, then it is very simple to show that φ is an isomorphism. This function, φ, shows that for any arbitrary group, G, there is one isomorphism from G to G but there are usually many others. Any of these functions - isomorphisms from a group onto itself - are called automorphisms, and in fact, the set of all possible automorphisms on a group forms a group itself under function composition.

Theorem: Aut(G) forms a group
Let G be any group and define Aut(G) to be the set of all automorphisms on G. Then Aut(G) forms a group under function composition.

Proof:
We denote the composition operation on Aut(G) by ∘.

Closure:

Choose φ,θ∈Aut(G). I wish to show that φ∘θ∈Aut(G) - that is that φ∘θ is an isomorphism from G to G. It is a property of bijections that a composition of bijections is also a bijection. I present this without proof because it is simple to verify. This gives that φ∘θ is a bijection, so we need to show that it is operation preserving. To see this, choose a,b∈G and observe that (φ∘θ)(a∙b) = φ(θ(a∙b)) = φ(θ(a)∙θ(b)) = φ(θ(a))∙φ(θ(b)) = (φ∘θ)(a)∙(φ∘θ)(b) so φ∘θ is operation preserving. Finally, this gives us that φ∘θ∈Aut(G) and Aut(G) is closed under function composition.

Associativity:

Everything in Aut(G) is a function and it is a property of all functions that their composition is associative.

Identity:

Define the function ε:G→G by ε(g)=g ∀g∈G. Now choose φ∈Aut(G) and choose g∈G. Observe that (ε∘φ)(g) = ε(φ(g)) = φ(g) and (φ∘ε)(g) = φ(ε(g)) = φ(g). It then follows that ε∘φ = φ = φ∘ε and ε is the identity element of Aut(G).

Inverses:

Choose φ∈Aut(G). Since φ is an isomorphism from G to G it folows from the first theorem in this post that φ^-1 is also an isomorphism from G to G and so φ^-1∈Aut(G) and by definition we have that φ∘φ^-1 = ε = φ^-1∘φ.

The above 4 properties give us that Aut(G) is a group under the operation of function composition.

I gave the proof because I think its worth reading. Usually I say that you can skip the proof if you'd like, but in this case I think the proof - though not particularly interesting - does give a good idea of the mechanism of the group and how it works. Inside of this group of automorphisms, there are some even more interesting functions called inner automorphisms. I'm not going to go into great detail about these and I'm not going to explain everything. They're sort of strange to understand, the proofs are tedious and boring, and I don't plan on talking about them for a long time, but whenever you're talking about the group of automorphisms, the group of inner automorphisms will probably be in the discussion so its worth defining.

Definition: Inner Automorphism Induced by a
Let G be a group and let a∈G. Then the function φ_a∈Aut(G) defined by φ_a(g) = a∙g∙a^-1 ∀g∈G is called the inner automorphism of G induced by a.

One thing to note is that this is a definition but it requires a bit of proof - that is, it is not obvious that φ_a∈Aut(G). However, it is a pretty easy thing to show.

Theorem: Inn(G) is a Subgroup of Aut(G)
Let Inn(G) be the set of all inner automorphisms on G. That is, if a∈G and φ_a is the inner automorphism induced by a, then Inn(G) = {φ_a : a∈G}. Then Inn(G) is a subgroup of Aut(G).

This theorem can be easily proven by a simple application of a subgroup test.

References

Previous Related Post: Isomorphisms

Text Reference: Gallain Chapter 6