Structure-preserving maps between groups -- kernels, images, and isomorphisms
Some maps between groups keep the multiplication intact. They show when two groups are secretly the same.
Choose a map, such as Z6 -> Z3. Elements of the source that land on the same target share a color.
Elements with the same color in the source map to the same target element.
A homomorphism is a function between groups that preserves the group operation: f(a * b) = f(a) * f(b). Homomorphisms are the "legal" maps between groups -- they respect structure. When a homomorphism is also a bijection, it is an isomorphism, meaning the two groups are structurally identical.
See two groups side-by-side with the mapping shown by coloring. Elements in the source that map to the same target share the same color.
Key insight: The map Z6 -> Z3 sends each element to its remainder mod 3. This preserves addition: (a+b) mod 3 = (a mod 3 + b mod 3) mod 3.
Choose a map and see which elements land on the identity. Together they form the kernel.
Kernel = elements that map to identity = {e, (0 3)(1 4)(2 5)}
The kernel is always a normal subgroup.
The kernel of a homomorphism is the set of elements that map to the identity. It is always a normal subgroup -- and the First Isomorphism Theorem says G/ker(f) is isomorphic to the image.
Key insight: The kernel measures how much information the homomorphism "forgets." A trivial kernel (just the identity) means the map is injective. The entire group as kernel means everything maps to the identity.
Pick two groups to see whether their tables match after relabeling. Try Z4 against V4: same size, yet not isomorphic.
Select two groups and check if they are isomorphic (have the same structure).
Same order (6) but not isomorphic. Different structure!
Two groups are isomorphic if there exists a structure-preserving bijection between them. Select two groups and check if they have the same Cayley table structure.
Key insight: Isomorphic groups are "the same group in disguise." Z4 and V4 both have order 4, but they are NOT isomorphic: Z4 has an element of order 4, while every non-identity element of V4 has order 2.
Spot a mistake or have an idea?