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.