Group

A group is a set with a well-behaved way to combine its elements. It is the minimal structure that captures "symmetry".

Definition

A group is a set G together with a binary operation · : G × G → G satisfying three axioms:

1. Associativity. (a · b) · c = a · (b · c) for all a, b, c ∈ G.
2. Identity. There is an element e ∈ G with e · a = a · e = a for every a ∈ G.
3. Inverses. For every a ∈ G there is an element a⁻¹ ∈ G with a · a⁻¹ = a⁻¹ · a = e.

The group is finite if the set G is finite; the number of elements is the order of the group. When context allows, we write ab for a · b and 1 for e.

A group in which additionally a · b = b · a for all a, b ∈ G is called abelian.

Intuition

Groups are what remain when you strip "symmetry" down to its essentials. Every kind of symmetry you can think of — rotations of a square, permutations of a deck of cards, invertible changes of coordinates — comes with three properties:

• You can compose two symmetries to get another symmetry.
• "Doing nothing" is a symmetry.
• Every symmetry can be undone.

The group axioms are exactly these three properties, nothing more. Every finite group is literally the symmetry group of something — this is Cayley's theorem.

Examples

• Cyclic group C₄ — rotations of a square by 0°, 90°, 180°, 270° under composition. 4 elements, abelian.
• Symmetric group S₃ — all 6 bijections of {1, 2, 3} under composition. Non-abelian (the smallest non-abelian group).
• Dihedral group D₄ — the 8 symmetries of a square: 4 rotations and 4 reflections. Non-abelian.
• Klein four-group V — the 4-element abelian group with every non-identity element of order 2.

Non-examples

• (ℕ, +) — the natural numbers under addition. Fails inverses: 3 has no additive inverse inside ℕ.
• A semigroup has only associativity; a monoid has associativity and identity. Neither is a group.

Properties

• A group has a unique identity element. If e and e' both act as an identity, then e = e · e' = e'.
• Every element of a group has a unique inverse. If b and b' are both inverses of a, then b = b · e = b · (a · b') = (b · a) · b' = e · b' = b'.
• Group multiplication cancels. In any group, ax = ay implies x = y, and xa = ya implies x = y.
• Inverting a product reverses the factors: (ab)⁻¹ = b⁻¹a⁻¹ in any group.

See also

• Order — the size of a group, the order of an element.
• Subgroup — subsets closed under the group operation.
• Abelian group — the commutative case.
• Cayley's theorem — every finite group is a permutation group.
• Wikipedia: Group (mathematics)