The definition, stated cleanly
A group is a set G together with a binary operation ∗ that satisfies the four group axioms. That's the whole definition — no numbers required, no geometry, just a set and a well-behaved way to combine its members. The four axioms are the four rules from the previous guide, now promoted to *requirements*.
- Closure: for all a, b in G, the result a ∗ b lies in G.
- Associativity: (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c in G.
- [[identity-element-of-a-group|Identity]]: there exists e in G with e ∗ a = a ∗ e = a for every a.
- [[inverse-element|Inverses]]: for each a in G there exists a⁻¹ in G with a ∗ a⁻¹ = a⁻¹ ∗ a = e.
Abelian and cyclic: two friendly families
If the operation also commutes — a ∗ b = b ∗ a for all a, b — the group is [[abelian-group|abelian]], named for Niels Henrik Abel. The integers under addition are abelian; so is addition mod n. Plenty of important groups are *not* abelian, like the symmetries of a triangle, where the order of moves changes the outcome.
A group is [[cyclic-group|cyclic]] if a single element generates the whole thing: combining one chosen element with itself, over and over, eventually produces *every* member. Addition mod 4 is cyclic because 1 generates it — 1, 1⊕1=2, 1⊕1⊕1=3, then back to 0. Every cyclic group is automatically abelian (repeating one element can't introduce any ordering conflict), but the reverse fails: not every abelian group is cyclic.
The order of an element
Pick an element and keep combining it with itself. In a finite group you must eventually land back on the identity. The number of steps that takes is the [[order-of-an-element|order of the element]] — the smallest positive integer k with aᵏ = e (here aᵏ means a combined with itself k times). The order of the identity is always 1. Computing a few orders tells you a lot about a group's shape.
Orders in (Z/6Z, ⊕), addition mod 6, identity e = 0 Here a^k means k copies of a added: a^k = (k·a) mod 6. element 1: 1, 2, 3, 4, 5, 0 → first hits 0 at k=6 order 6 element 2: 2, 4, 0 → first hits 0 at k=3 order 3 element 3: 3, 0 → first hits 0 at k=2 order 2 element 4: 4, 2, 0 (4,8mod6=2,12mod6=0) → k=3 order 3 element 5: 5,4,3,2,1,0 → k=6 order 6 element 0: already e order 1 Notice: every order (1,2,3,6) divides the group size 6. Also: elements 1 and 5 each have order 6 = |G|, so each one generates the whole group — it is CYCLIC.