What Is an Isomorphism?

In mathematics, an isomorphism is a function that shows two mathematical objects have the same structure. Although the objects may look different, an isomorphism demonstrates that they behave in exactly the same way with respect to the operations that define them. If such a map exists, the objects are called isomorphic.

An isomorphism tells us that two systems are essentially the same, differing only by a relabelling of their elements.

The Basic Idea

An isomorphism is a function:

f : A → B

that must be:

• Injective — different elements of A map to different elements of B.
• Surjective — every element of B comes from some element of A.

Together these mean f is bijective, and so it has an inverse:

f⁻¹ : B → A

No information is lost moving from A to B or back.

Preserving Structure

Bijectivity alone is not enough. An isomorphism must also preserve structure. For groups, this means:

f(a ⋆ b) = f(a) ∘ f(b)

for all a, b ∈ A, where ⋆ is the operation in A and ∘ is the operation in B. Performing the operation first or mapping first must give the same result.

Example: ℤ₄ and the 4th Roots of Unity

Consider the group of integers modulo 4 under addition:

ℤ₄ = {0, 1, 2, 3}

Its operation table is:

+	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

Now consider the group of 4th roots of unity under multiplication:

U = {1, i, −1, −i}

×	1	i	−1	−i
1	1	i	−1	−i
i	i	−1	−i	1
−1	−1	−i	1	i
−i	−i	1	i	−1

Define a function:

f(0) = 1
f(1) = i
f(2) = −1
f(3) = −i

This mapping is clearly bijective. To verify it is an isomorphism, we must check:

f(a + b mod 4) = f(a) × f(b)

for all a, b. For example:

a = 1, b = 1

Left side:  f(1 + 1 mod 4) = f(2) = −1
Right side: f(1) × f(1) = i × i = −1

Both sides agree. Checking all pairs confirms that the group operation is preserved, and so ℤ₄ and U are isomorphic. They behave identically despite appearing different.

Why Is This Important?

When two objects are isomorphic, they can be treated as essentially the same. Results proven for one apply to the other without modification. Isomorphisms allow us to transfer problems to simpler settings, solve them there, and translate solutions back.

Other Settings

Isomorphisms occur throughout mathematics:

• Vector spaces: preserve vector addition and scalar multiplication.
• Rings: preserve addition and multiplication.
• Topological spaces: homeomorphisms preserve open sets.
• Metric spaces: isometries preserve distances.

Another Example: Vector Space Isomorphism

Consider ℝ² and the set of 2×1 real column vectors:

V = { (a, b)ᵀ }

Define:

f((x, y)) = (x, y)ᵀ

This mapping is bijective, and it preserves vector addition and scalar multiplication. Thus, ℝ² and V are isomorphic as vector spaces — same structure, different presentation.

Summary

• An isomorphism is a bijection that preserves structure.
• Two isomorphic objects are structurally identical.
• Isomorphisms let us translate problems into simpler settings.

f : A → B   is an isomorphism
⇔
f is bijective and preserves structure