THE GROUP ACTION — a dart

THE GROUP ACTION how symmetries compose — do one then another, land as if you did their product

A group action is how a group of symmetries actually moves a thing, consistently: doing transformation h then g lands you exactly where the single transformation g·h would — g·(h·x) = (g·h)·x — and the identity does nothing. It is the bridge from the abstract group (a multiplication table) to its effect on a set. Rotate a square by 90° then 180° and you have rotated it 270°; the actions compose the way the rotations do. Every symmetry in the batch — reflections, rotations, permutations, shifts — is a group acting, and this compatibility is what makes “symmetry” a coherent structure rather than a loose collection of moves.

THE TECHNIQUE g·(h·x) = (g·h)·x ; identity fixes x

The demo rotates a list by 1, then by 2, and checks it equals rotating by 3 — the actions compose: live demo

HISTORY & CREDIT Galois · Cayley · Klein

“Symmetries are a bag of moves.” — they compose lawfully: do two in a row and you land where their product would. A group acting is a structure, not a list. cited

the axioms · (g·h)·x = g·(h·x) and e·x = x — the action respects the group.
Cayley · every group acts on itself; Klein's Erlangen program — geometry is the study of a group acting.
the batch · reflections, rotations, permutations, shifts — all group actions.

Do one symmetry then another and land where their product would — the moves obey the group. Symmetry as a lawful structure. Galois / Klein

RECOMMEND FOR I-13 rotations composing, on the compiler

On the canonical compiler, rotating [1,2,3,4] by 1 then 2 equals rotating by 3 (compose=1):

$ i13 run sy_groupaction.i13 # rot(rot(x,1),2) == rot(x,3) RUN OK · 473 step(s) · peak stack 8 · call depth 5 compose = 1 -- the actions compose the way the rotations do

Recommend: the group action is what makes symmetry a structure — the moves compose lawfully — and i13 confirms rotate-1-then-2 equals rotate-3. Not a keeper (associativity of an action is a group axiom, witnessed). The dart that underwrites the whole batch: reflections, rotations, permutations, shifts are all one thing — a group acting — which is why equivariance to them is even well-defined.