A group is a set equipped with an operation that combines any two elements to produce a third, subject to four rules:

1. Closure: Combining any two elements gives an element in the set
2. Associativity: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
3. Identity: There’s an element $e$ such that $e \cdot a = a \cdot e = a$ for all $a$
4. Inverses: Every element $a$ has an inverse $a^{-1}$ with $a \cdot a^{-1} = e$

That’s it. These four axioms capture the essence of symmetry — and they appear everywhere.

Permutation groups

The most concrete groups are permutation groups — groups of rearrangements. If you’ve worked through permutations, composing permutations, and cycle notation, you already know the key facts:

• The set of all permutations of $n$ objects forms the symmetric group $S_n$, with $n!$ elements
• Composition of any two permutations gives another permutation (closure)
• Composition is associative but not commutative
• The identity permutation leaves everything in place
• Every permutation has an inverse (undo the rearrangement)

That’s all four group axioms, verified. $S_n$ is a group — and a rich one.

Subgroups

A subgroup is a subset of a group that is itself a group under the same operation. For example, ${e, (12)}$ is a subgroup of $S_3$ — it contains the identity, and $(12)$ is its own inverse.

The order of a group is the number of elements. Lagrange’s theorem says: the order of any subgroup divides the order of the group. $S_3$ has order 6, so its subgroups can have order 1, 2, 3, or 6.

Why groups matter

Groups capture the abstract structure of symmetry: - Rotations and reflections of a square form the dihedral group $D_4$ (8 elements) - Invertible matrices form groups under multiplication - The integers $\mathbb{Z}$ form a group under addition - Permutations of roots of a polynomial form the Galois group — the key to understanding when equations can be solved