Group Homomorphisms

Group Homomorphisms

A group homomorphism is a function $\phi: G \to H$ between groups that preserves the group operation:

$\phi(a *_G b) = \phi(a) *_H \phi(b)$

Properties of Homomorphisms

If $\phi: G \to H$ is a homomorphism, then:

• $\phi(e_G) = e_H$ (identity maps to identity)
• $\phi(a^{-1}) = \phi(a)^{-1}$ (inverses map to inverses)

Important Types

• Isomorphism: bijective homomorphism (groups are "the same" structurally)
• Automorphism: isomorphism from a group to itself
• Kernel: $\ker(\phi) = \{a \in G : \phi(a) = e_H\}$ — always a subgroup of $G$

Example: $\phi: \mathbb{Z}_6 \to \mathbb{Z}_3$

Define $\phi(a) = a \bmod 3$. This is a homomorphism because: $(a + b) \bmod 3 = ((a \bmod 3) + (b \bmod 3)) \bmod 3$

The kernel is $\ker(\phi) = \{0, 3\}$ — elements that map to $0$ in $\mathbb{Z}_3$.

Checking Homomorphism Computationally

Given groups $\mathbb{Z}_m$ and $\mathbb{Z}_n$ and a function $\phi$, verify:

def is_homomorphism(phi, m, n):
    for a in range(m):
        for b in range(m):
            if phi((a + b) % m) != (phi(a) + phi(b)) % n:
                return False
    return True

Your Task

Implement is_homomorphism(phi, m, n) that checks if function phi (a list where phi[a] gives $\phi(a)$) is a homomorphism from $\mathbb{Z}_m \to \mathbb{Z}_n$ under addition, and kernel(phi, m) that returns the sorted kernel of $\phi$.