Algebra 9 | Group Homomorphisms

TL;DR

A group homomorphism F: G → H preserves the group operation via F(a · b) = F(a) ∗ F(b) for all a, b in G.

Briefing Cornell Notes

Briefing

Group homomorphisms are the structure-preserving maps between two groups: they must respect the group operation in a way that makes “apply the map, then combine” match “combine first, then apply the map.” Concretely, for groups (G, ·) and (H, ∗), a function F: G → H is a group homomorphism if for all a, b in G,

F(a · b) = F(a) ∗ F(b).

That single equation is enough to guarantee the identity and inverse behavior usually associated with homomorphisms. In other words, once the operation is preserved, the map automatically sends the identity element of G to the identity element of H, and it also carries inverses correctly.

The transcript then builds intuition with a classic example that translates addition into multiplication. Take G as the real numbers under addition, and H as the nonzero real numbers under multiplication (so 0 is excluded to keep a multiplicative identity and inverses). Define F(x) = e^x. This map lands in H because e^x is never zero. Checking the homomorphism property reduces to the exponent rule: for x and y,

F(x + y) = e^(x+y) = e^x · e^y = F(x) ∗ F(y).

So the exponential function becomes a group homomorphism from (R, +) to (R\{0}, ·). The example also highlights an important point about surjectivity: a homomorphism does not have to hit every element of the target group. Even if F is not onto, it still embeds a whole copy of the source group inside the target in a way compatible with the group structure.

After establishing the definition and example, the transcript proves the “extra” properties that were not explicitly required. First, it shows that the identity in G must map to the identity in H. Using the homomorphism equation with the identity element e_G, the calculation forces F(e_G) to behave as the identity under the operation in H, so F(e_G) = e_H.

Second, it proves the inverse rule. Since every element a in a group can be written using its inverse (e.g., a^(-1) · a = e_G), applying the homomorphism property transfers that relationship into H. The uniqueness of inverses in a group then implies that

F(a^(-1)) = (F(a))^(-1).

The overall takeaway is practical: once a map preserves the group operation, it automatically preserves the identity and inverses too, which makes algebraic manipulations cleaner—especially when pulling inverse signs or combining elements through the homomorphism. The transcript closes by noting that further applications of homomorphisms appear in later lessons.

Cornell Notes

A group homomorphism is a function F: G → H between groups that preserves the group operation: F(a · b) = F(a) ∗ F(b) for all a, b in G. That single condition is powerful enough to force two additional facts: the identity element of G maps to the identity element of H, and inverses are preserved in the sense that F(a^(-1)) = (F(a))^(-1). A key example uses the exponential function: F(x) = e^x is a homomorphism from (R, +) to (R\{0}, ·) because e^(x+y) = e^x e^y. Homomorphisms also need not be surjective; they can embed the structure of G into H without covering every element of H.

Why does preserving only the group operation already guarantee identity and inverse preservation?

Because the homomorphism condition F(a · b) = F(a) ∗ F(b) can be applied using special choices of a and b. For the identity, set a = e_G and use that e_G · a = a; this forces F(e_G) to act as the identity under ∗ in H. For inverses, use the fact that a^(-1) · a = e_G; applying F and the homomorphism equation transfers this to H, and uniqueness of inverses in a group implies F(a^(-1)) = (F(a))^(-1).

How does the exponential function become a group homomorphism?

Let G = (R, +) and H = (R\{0}, ·). Define F(x) = e^x. Then for x, y in R: F(x + y) = e^(x+y) = e^x · e^y = F(x) ∗ F(y). The exponent rule is exactly the homomorphism condition, and e^x never equals 0, so the map lands in H.

What does it mean that a homomorphism need not be surjective?

Surjectivity would mean every element of H is hit by some F(a). The transcript stresses that homomorphisms can fail to hit all of H, yet still preserve the group structure on the image. In that case, the image forms a subgroup-like copy of G inside H, even if some elements of H are never reached.

How can one prove that F(e_G) equals the identity e_H?

Use the homomorphism property with the identity element. Since e_G behaves neutrally under the operation in G, inserting e_G into the homomorphism equation forces F(e_G) to behave neutrally under the operation in H. The calculation shows that F(e_G) must satisfy the defining property of the identity in H, so F(e_G) = e_H.

What is the inverse rule for homomorphisms, and why is it useful?

The inverse rule is F(a^(-1)) = (F(a))^(-1). It follows from applying the homomorphism equation to a^(-1) · a = e_G and using uniqueness of inverses in H. It’s useful because it lets algebraic expressions involving inverses be “pulled through” the homomorphism, simplifying computations.

Review Questions

State the definition of a group homomorphism and write the equation it must satisfy for all a, b in G.
Give the groups used in the exponential example and verify the homomorphism property using an exponent rule.
Explain why a homomorphism must send the identity element of G to the identity element of H.

Key Points

1
A group homomorphism F: G → H preserves the group operation via F(a · b) = F(a) ∗ F(b) for all a, b in G.
2
Preserving the operation automatically forces F(e_G) = e_H, so identity preservation need not be added separately.
3
A group homomorphism also preserves inverses: F(a^(-1)) = (F(a))^(-1).
4
The exponential function F(x) = e^x is a homomorphism from (R, +) to (R\{0}, ·) because e^(x+y) = e^x e^y.
5
Homomorphisms do not have to be surjective; they can embed the structure of G into H without covering every element of H.
6
When working with homomorphisms, inverses and combinations can often be transferred through F, simplifying group calculations.

Highlights

A single equation—F(a · b) = F(a) ∗ F(b)—is enough to guarantee both identity and inverse preservation.

F(x) = e^x turns addition on real numbers into multiplication on nonzero real numbers, matching the exponent rule exactly.

Even without hitting every element of the target group, a homomorphism still preserves the group structure on its image.

Topics

Group Homomorphisms
Homomorphism Properties
Exponential Example
Identity and Inverses