02 / Identity
Same in what sense?
Write two objects.
Discover what a map preserves.
Comparing abstract groups
Example result · change the inputs or compare againℤ/4ℤ / C₄
Yes — they are isomorphic as abstract groups.
ℤ/4ℤ describes four residue classes under addition, while C₄ describes four rotations. Pairing [k] with the rotation through 90k° preserves the group operation, so the two objects have the same abstract group structure.
Explicit map
f([k]) = rᵏ
Here r is a rotation through 90°. Select a pair to see the operation in both representations.
f([1] + [2]) = f([1]) · f([2]) = r³
Left: addition modulo 4. Right: composition of rotations.
Show mathematical verification
Definitions & evidence
✓ Computedℤ/4ℤ
Residue classes · addition modulo 4
C₄
Cyclic group · r^4 = 1 · powers of a fixed rotation
What matches?
- Both have 4 elements.
- Both are commutative.
- They have the same element-order pattern.
These are invariants — properties that a group isomorphism must preserve. Matching properties alone do not prove an isomorphism.
Isomorphic finite groups have the same number of elements. The converse is false: C₄ and C₂ × C₂ both have order 4.
Why the map works
The map pairs every element exactly once, and all 16/16 operation checks pass.
Closure, associativity, identity and inverses are checked on both groups. A separate verification of the proposed map confirms the bijection and operation preservation. Preservation of identity, inverses, element orders and subgroups follows from the isomorphism. The map is one possible choice; it need not be unique or canonical.
Inspect both operation tables
ℤ/4ℤ
| + | [0] | [1] | [2] | [3] |
|---|---|---|---|---|
| [0] | [0] | [1] | [2] | [3] |
| [1] | [1] | [2] | [3] | [0] |
| [2] | [2] | [3] | [0] | [1] |
| [3] | [3] | [0] | [1] | [2] |
C₄
| · | 1 | r | r² | r³ |
|---|---|---|---|---|
| 1 | 1 | r | r² | r³ |
| r | r | r² | r³ | 1 |
| r² | r² | r³ | 1 | r |
| r³ | r³ | 1 | r | r² |
Group structure & smooth structure
- Abstract groups — preserve the operation
- Isomorphic. A bijection preserving every operation pair was found and independently checked.
- Lie groups — also preserve smooth structure
- Isomorphic. A bijection preserving every operation pair was found and independently checked. Both groups carry the discrete topology, so every group isomorphism is also a Lie-group isomorphism.
- Real Lie algebras — compare infinitesimal structure
- Isomorphic. Both Lie algebras are zero. In particular every finite group with the discrete topology has this same Lie algebra; it cannot distinguish finite groups.
Method & references
Finite-group results use exact operation tables, group-axiom checks and a bounded search with independently verified witnesses. A search limit yields “not determined”. Continuous-group facts and special maps are sourced theorems; numerical diagrams illustrate them. No AI generates these verdicts.
- J. S. Milne · Group Theory ↗
Cyclic groups, homomorphisms and direct products: residue-to-rotation maps and products of cyclic groups of coprime orders.
- Anthony W. Knapp · Lie Groups Beyond an Introduction ↗
Introduction and Chapter I, §§11 and 17: classical matrix groups, real Lie algebras, compact forms and covering groups. Sp(n) uses the compact quaternionic convention.
- Vincent Bouchard · SU(2) ↗
SU(2) and SO(3): isomorphic real Lie algebras, the two-to-one covering homomorphism, and the global distinction between the groups.
- Erich Reck & Georg Schiemer · Structuralism in the Philosophy of Mathematics ↗
Structuralist interpretations, identity and representation.
If two objects look the same nearby, what can still distinguish them globally?
Try SU(2) and SO(3), then U(1) and SO(2). Compare the maps, their kernels and the structures each map preserves.