4. Symmetries and Groups

Central intuition

A symmetry of an object is a reversible transformation that may change how the object is presented, but preserves the structure we care about.

Informally:

A symmetry changes the view, not the essence.

Groups are the natural language of symmetries because symmetries can be composed, have an identity transformation, and can be undone.

What does it mean to preserve structure?

It depends on the object.

For a geometric figure, preserving structure may mean preserving distances and angles.

For a graph, it means preserving adjacency.

For an algebraic structure, it means preserving operations.

So the word “symmetry” always comes with the question:

What features of the object count as essential?

Example: symmetries of a square

A square has eight rigid symmetries:

• rotation by $0^\circ$;
• rotation by $90^\circ$;
• rotation by $180^\circ$;
• rotation by $270^\circ$;
• four reflections.

These transformations form a group because:

1. composing two symmetries gives another symmetry;
2. doing nothing is a symmetry;
3. every symmetry can be undone;
4. composition of transformations is associative.

This group is called the dihedral group of the square.

Why symmetries must be reversible

If a transformation loses information, it is not a symmetry.

For example:

• rotating a square is reversible;
• reflecting a square is reversible;
• crushing a square into a line is not reversible;
• projecting a 3D object onto a plane usually loses information.

A symmetry is not just an action. It is an invertible action that preserves the relevant structure.

Formal viewpoint: automorphisms

In abstract mathematics, a symmetry of a structure is often called an automorphism.

An automorphism is a structure-preserving bijection from an object to itself.

Examples:

• a symmetry of a set is a bijection;
• a symmetry of a graph is a vertex permutation preserving edges;
• a symmetry of a group is a bijective homomorphism $G\to G$;
• a symmetry of a vector space is an invertible linear map.

The set of all automorphisms of an object forms a group.

Why symmetries matter

Symmetries reveal what is essential and what is arbitrary.

If an object has many symmetries, then many apparent differences are not structurally meaningful.

For example:

• all directions on a circle are equivalent under rotation;
• the vertices of a regular polygon are interchangeable by symmetry;
• coordinates in a vector space may change while the underlying vector remains the same.

A group of symmetries often acts like a fingerprint of the object.

Group actions

A group can act on a set $X$. This means each group element is realized as a symmetry or transformation of $X$.

Formally, a group action assigns to each $g\in G$ a function $X\to X$, satisfying

\(e\cdot x=x,\) and

\((gh)\cdot x=g\cdot(h\cdot x).\) This is how abstract groups become concrete transformations.

Place in the build-up

The idea of symmetry explains why groups are central.

A group is not just an arbitrary algebraic object. It is exactly what appears when we collect all reversible transformations preserving a structure.

This is why groups occur in geometry, number theory, physics, combinatorics, and the theory of equations.