Symmetry operations and the group axioms

A binary operation takes two elements of a specified set and returns one element of that same set. For symmetries, the operation is composition. We write $ab$ for doing $b$ first and $a$ second, consistently with composition of functions. The set must be closed under that operation. Quarter turns alone are not closed: composing two quarter turns gives a half turn. Including all four rotations of the square repairs this problem.

Associativity says that regrouping three operations changes no result. It does not authorize changing their order. Function composition is associative because both parenthesizations apply the same three functions in the same sequence to every input. An identity operation leaves every input untouched. An inverse undoes a transformation on both sides. A symmetry is therefore a bijection of the underlying object preserving the structure we chose to care about, such as distances between corners.

Place a square at coordinates $(\pm1,\pm1)$. Let $r$ rotate counterclockwise through ninety degrees and let $s$ reflect across the horizontal axis. A corner at $(1,1)$ is sent to $(1,-1)$ by $s$ and then to $(1,1)$ by $r$, so $rs$ fixes that corner. Doing the rotation first sends it to $(-1,1)$; reflecting then gives $(-1,-1)$. Thus $rs$ and $sr$ are different transformations.

The eight distance-preserving symmetries of the square form a nonabelian group: some pairs fail to commute. In contrast, the four rotations commute because their angles add modulo a full turn. A group is called abelian when every pair commutes. Never insert commutativity into a proof merely because the symbols look like ordinary numbers. Matrix multiplication and permutations will repeatedly remind us that the notation hides the nature of the operation.

Suppose $ab=ac$. Multiplying both sides on the left by the inverse of $a$ and regrouping gives $b=c$. This is left cancellation; right cancellation follows by multiplying on the right. The side matters when multiplication is noncommutative. Similarly, if two elements both claim to be the identity, their product equals each of them, proving they are equal. The axioms therefore guarantee one identity rather than requiring an additional uniqueness assumption.

The inverse of a product reverses the order: to undo putting on socks and then shoes, remove the shoes first. Algebra verifies this without an analogy: multiply $ab$ by $b^{-1}a^{-1}$ and cancel adjacent inverse pairs. Both orders of multiplication give the identity. These short proofs are the first benefit of abstraction: once established, the same reasoning works for rotations, invertible matrices, and permutations without inspecting their particular representations.