Groups and Symmetry

Entry conditions

Use groups when you need to reason about operations that can be reversed or about the symmetries of a structure.

Definitions

A group is a monoid $(G, \cdot, e)$ in which every element has an inverse: for each $g \in G$ there exists $g^{-1} \in G$ such that $g \cdot g^{-1} = e = g^{-1} \cdot g$.

Unpacking, a group satisfies four properties:

1. Closure: if $a, b \in G$, then $a \cdot b \in G$.
2. Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
3. Identity: there exists $e$ such that $e \cdot a = a \cdot e = a$ for all $a$.
4. Inverses: for each $a$, there exists $a^{-1}$ with $a \cdot a^{-1} = e$.

If additionally $a \cdot b = b \cdot a$ for all $a, b$, the group is abelian.

Vocabulary (plain language)

• Group: a set with an associative operation, an identity, and inverses.
• Inverse: the element that undoes another.
• Abelian: a group where order of operation does not matter.

Symbols used

• $(G, \cdot, e)$: a group with operation $\cdot$ and identity $e$.
• $g^{-1}$: the inverse of $g$.

Intuition

A monoid lets you combine things; a group lets you also take them apart. The integers under addition form a group: you can always subtract. The natural numbers under addition do not, because you cannot subtract past zero.

Symmetry is the prototypical source of groups. The rotations of a square form a group of four elements. The rotations and reflections together form the dihedral group $D_4$ of eight elements. Every group can be realized as symmetries of some object — this is Cayley’s theorem.

Worked example

The integers $(\mathbb{Z}, +, 0)$ form an abelian group. For any integer $n$, its inverse is $-n$, since $n + (-n) = 0$. Associativity and commutativity of addition are inherited from arithmetic.

How to recognize the structure

• You have a binary operation that is associative.
• There is an identity element.
• Every element can be “undone.”

Common mistakes

• Confusing groups with monoids: monoids do not require inverses.
• Assuming all groups are abelian. Matrix multiplication gives non-abelian groups.
• Forgetting that the inverse must satisfy $g \cdot g^{-1} = e$ on both sides.