abelian group
An abelian group is a group whose operation does not care about order: combining a then b gives the same result as b then a. It is the “polite” kind of group, where everything commutes. Ordinary addition behaves this way — 3 + 5 and 5 + 3 are both 8 — which is exactly why arithmetic feels so forgiving.
Formally, a group G is abelian if a * b = b * a holds for every pair a, b in G. This single extra requirement, added on top of the four group axioms, has large consequences: the structure becomes far more rigid and far easier to classify. The name honours the Norwegian mathematician Niels Henrik Abel.
Not every group is abelian. The symmetries of a triangle and the act of shuffling cards are classic non-abelian examples, because order of operations genuinely changes the outcome. When a group's operation is written with a “+” sign, mathematicians almost always intend it to be abelian; multiplicative notation makes no such promise.
The integers (ℤ, +) are abelian: m + n = n + m always. But the symmetric group S₃ (all six rearrangements of three objects) is not — swapping items 1 and 2 then 2 and 3 differs from doing those swaps in the opposite order.
(ℤ, +) commutes; S₃ does not.