math / Structures / free

Group Theory

Use reversible operations to map the hidden symmetries of a system.

Follow the mechanism1 / 3

A shape admits structure-preserving actions. Rotations and reflections move a square's labels while preserving the relationships among its corners and sides.

square symmetryD₄ · r · order 4

Focus the idea

original labelsafter rcomposition stays inside D₄

Choose an action

Composition table · row then column
∘	e	r	r²	r³	s	rs	r²s	r³s
e	e	r	r²	r³	s	rs	r²s	r³s
r	r	r²	r³	e	r³s	s	rs	r²s
r²	r²	r³	e	r	r²s	r³s	s	rs
r³	r³	e	r	r²	rs	r²s	r³s	s
s	s	rs	r²s	r³s	e	r	r²	r³
rs	rs	r²s	r³s	s	r³	e	r	r²
r²s	r²s	r³s	s	rs	r²	r³	e	r
r³s	r³s	s	rs	r²s	r	r²	r³	e

symmetries8

identitye

inverser³

order of action4

Selected actionr

rotate 90°.

Undo it withr³

r ∘ r³ = e.

Repeat itr²

r ∘ r = r²; after 4 repeats, the square returns to its start.

r is a reversible action: it rearranges the four vertices without changing the square. Its inverse is r³, which brings every label back.

Group theory studies reversible actions as objects that can be composed. The square is a compact example: four rotations and four reflections form one closed system, so a sequence of moves can be simplified, undone, and compared structurally.