# 4.5 Examples: Groups, Spaces, Graphs

## 4.5 Examples: Groups, Spaces, Graphs

Structural thinking becomes clearer through examples. This section shows how the same principles appear in different areas.

We focus on three domains: groups, topological spaces, and graphs. Each has its own objects, morphisms, invariants, and classification problems.
### 1. Groups

A group is a set $G$ with a binary operation satisfying associativity, identity, and inverses.

#### Objects and maps

* Objects: groups $(G, \ast)$

* Morphisms: group homomorphisms
  $$
  f(a \ast b) = f(a) \cdot f(b)
  $$

* Isomorphisms: bijective homomorphisms

#### Example instances

| Group                        | Description                            |
| ---------------------------- | -------------------------------------- |
| $(\mathbb{Z}, +)$            | Integers under addition                |
| $(\mathbb{Q}^\times, \cdot)$ | Nonzero rationals under multiplication |
| $S_n$                        | Permutations of $n$ elements           |
| $D_4$                        | Symmetries of a square                 |

These are different representations of the same structure type.

#### Invariants

| Invariant          | Meaning               |   |                                  |
| ------------------ | --------------------- | - | -------------------------------- |
| Order $            | G                     | $ | Number of elements (finite case) |
| Abelian property   | Whether $ab = ba$     |   |                                  |
| Subgroup structure | Internal organization |   |                                  |
| Element orders     | Cyclic behavior       |   |                                  |

#### Example classification

* Every group of prime order $p$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$
* Finite abelian groups can be classified by decomposition into cyclic factors

This shows structural classification via invariants.
### 2. Topological spaces

A topological space is a set $X$ together with a collection of open sets.

#### Objects and maps

* Objects: spaces $(X, \mathcal{T})$
* Morphisms: continuous maps
* Isomorphisms: homeomorphisms

A map $f : X \to Y$ is continuous if

$$
f^{-1}(U) \text{ is open in } X \quad \text{for every open } U \subseteq Y.
$$

#### Example instances

| Space          | Description                               |
| -------------- | ----------------------------------------- |
| $\mathbb{R}$   | Real line with usual topology             |
| $[0,1]$        | Closed interval                           |
| Circle $S^1$   | Points at unit distance in $\mathbb{R}^2$ |
| Discrete space | Every subset is open                      |

#### Invariants

| Invariant            | Meaning                                 |
| -------------------- | --------------------------------------- |
| Connectedness        | Cannot be split into disjoint open sets |
| Compactness          | Finite subcover property                |
| Number of components | Decomposition into pieces               |
| Fundamental group    | Loop structure (advanced)               |

#### Example comparison

* $[0,1]$ and $(0,1)$ are not homeomorphic
* $\mathbb{R}$ and $(0,1)$ are homeomorphic

This shows that topology ignores distance but preserves open-set structure.
### 3. Graphs

A graph consists of vertices and edges.

#### Objects and maps

* Objects: graphs $G = (V, E)$
* Morphisms: graph homomorphisms
* Isomorphisms: bijections preserving adjacency

#### Example instances

| Graph                | Description             |
| -------------------- | ----------------------- |
| Path graph           | Chain of vertices       |
| Cycle graph          | Closed loop             |
| Complete graph $K_n$ | All pairs connected     |
| Tree                 | Connected acyclic graph |

#### Invariants

| Invariant            | Meaning                   |
| -------------------- | ------------------------- |
| Number of vertices   | Size                      |
| Number of edges      | Connectivity level        |
| Degree sequence      | Vertex degrees            |
| Connected components | Structure of connectivity |
| Presence of cycles   | Loop structure            |

#### Example comparison

Two graphs with different degree sequences cannot be isomorphic.
However, graphs with the same degree sequence may still be non-isomorphic.

This shows invariants can be incomplete.
### 4. Cross-domain comparison

The same structural pattern appears in all three domains.

| Concept        | Groups              | Topology                   | Graphs              |
| -------------- | ------------------- | -------------------------- | ------------------- |
| Objects        | Groups              | Spaces                     | Graphs              |
| Morphisms      | Homomorphisms       | Continuous maps            | Graph homomorphisms |
| Isomorphism    | Group isomorphism   | Homeomorphism              | Graph isomorphism   |
| Invariants     | Order, abelian      | Compactness, connectedness | Degree, components  |
| Classification | Partial or complete | Often partial              | Often hard          |

This table shows that structural thinking is uniform across fields.
### 5. Representation vs structure

Each domain has multiple representations.

| Object | Representation       | Structure          |
| ------ | -------------------- | ------------------ |
| Group  | Multiplication table | Abstract operation |
| Space  | Coordinates          | Open sets          |
| Graph  | Adjacency list       | Connectivity       |

Different representations can describe the same structure.

For example, a graph can be stored as:

* adjacency list
* adjacency matrix
* edge set

These differ in representation but encode the same structure.
### 6. Key lessons

* Structure defines what matters
* Morphisms define allowed transformations
* Isomorphism defines sameness
* Invariants detect differences
* Classification organizes objects

These ideas repeat across mathematics.

A problem in one domain often has an analogue in another. Structural thinking allows results to transfer.

For example:

* Decomposition in groups parallels decomposition in vector spaces
* Connectivity in graphs parallels connectedness in topology
* Symmetry in geometry parallels automorphisms in algebra
### 7. Practical workflow

When facing a new structure:

| Step | Action                 |
| ---- | ---------------------- |
| 1    | Identify objects       |
| 2    | Identify morphisms     |
| 3    | Determine invariants   |
| 4    | Compare via invariants |
| 5    | Attempt classification |

This workflow applies across domains.
Structural thinking becomes effective when you see the same pattern in different contexts. Groups, spaces, and graphs look different on the surface, but they follow the same underlying design.