the First Isomorphism Theorem, expanded

matan-matika:

In a previous post, we proved the first isomorphism theorem in the form “If φ:G→H is a group homomorphism, then G/Ker(φ) ≅ Im(φ)”

We can restate the theorem, and get some fascinating results.

The theorem states that there exists an isomorphism I:G/Ker(φ) → Im(φ).
Now, Im(φ) is a subgroup of H, so we can make function i:G/Ker(φ) → H out of I with all the same values, just a different domain.

i inherits lots of properties from I, just doesn’t go the parts of H that are not in Im(φ). Or, the property we lose is surjectivity.

It still preserves structure (it’s a homomorphism), and it keeps the same 1-to-1 mapping (injectivity).

An injective homomorphism is just saying “take a smaller group, and put it exactly the same in a larger group”

So, we can really restate this theorem as “if φ:G→H is a group homomorphism, then you can do the same thing as φ by quotienting, and then injecting the quotient into H.”

Let’s try it in mathier terms: Let’s say φ:G→H is a group homomorphism, and q:G→G/Ker(φ) is the quotient map, then there exists an injective homomorphism i:G/Ker(φ)→H such that i∘q=φ

Note that function composition does the right most thing first. Also note that even though we still call it “the first isomorphism theorem,” this statement of it doesn’t have any isomorphisms in it.

Here’s a nice diagram of the theorem.

A solid line means “assumed (like φ) or already proven from assumption (like q),” and a dashed line means “the theorem states that this exists (like i)“

If you see the “φ=i∘q” version, you might realize that this theorem is stating:
“For any group homomorphism φ, you can factor φ into a part that crunches it down, q, and a part that puts the crunched down part into the target group, i.”

Here’s a homomorphism from the Klein 4 group, which we denote V, toanother copy of V.

Here is it factored into a quotient and an injection.