Teaching objectives
First lab of the group theory series, for upper secondary school or first-year university. It is followed by “Cosets and quotients”, “Homomorphisms and automorphisms” and “Centre, commutators and solvable groups”.
A group does not start here with axioms, but with actions that leave an object in place: the phone that goes back into its spot in four ways, the triangle in its frame, the spinning tetrahedron. Each position is also an action (the one that takes you from the start to it), and chaining actions is multiplying. Then the same group is drawn in three ways (Cayley diagram, coloured multiplication table and cycle graph) and students learn to recognise it even when the picture changes.
The convention is Carter’s: ab means “first a, then b”, which is following the arrows in the diagram and reading a Rubik’s cube formula from left to right.
What students learn
- To recognise a group in the symmetries of an object and name its elements with words in the generators (r, f, r²f…).
- The four rules of a group (fixed actions that can be undone, are predictable and can be chained) and how they translate into textbook axioms.
- To build and read Cayley diagrams: following words, seeing that two paths to the same node are an equation, changing generators and spotting impossible diagrams.
- To fill in and read the multiplication table: the sudoku property, the identity, inverses, and symmetry about the diagonal when the group is abelian.
- To compute the order of an element and read it off the cycle graph.
- To work with permutations: cycle notation, composition, parity, Sₙ and Aₙ.
- To decide whether two groups are isomorphic by comparing invariants (order, abelian, distribution of orders).
- To count the rotations of the tetrahedron, cube and icosahedron by type of axis and recognise A₄, S₄ and A₅.
- To see 3D Cayley diagrams as orbits of a point and read relations between generators off their faces.
- To compute the order of 2×2×2 Rubik’s cube formulas.
Key mathematical ideas
- A position is an action. What matters about an action is where it leaves the object, not how it moved along the way.
- Order matters, sometimes. For the phone hv = vh; for the square rf ≠ fr = r³f.
- Two paths, one node, one equation. Relations such as r³ = e or fr = r²f show up as closed paths in the diagram.
- Each row of the table contains each element once, because every action can be undone.
- Invariants decide isomorphism for small groups. The phone is not Z₄ (it has no elements of order 4); Q₈ is not Δ₄ (1 versus 5 elements of order 2).
- The regular polyhedra hide A₄, S₄ and A₅. The 24 rotations of the cube rearrange its 4 diagonals in all 24 possible ways.
Sources and credits
The diagrams, the four rules and the idea “position = action” follow Visual Group Theory by Nathan Carter (MAA, 2009). The problem-based approach and the rotation groups of the polyhedra follow Abel’s Theorem in Problems and Solutions by V. B. Alekseev, based on V. I. Arnold’s lectures. No Group Explorer code is used.
Classroom notes
- “The phone in its case” can be done with a real phone before opening the lab: in how many ways does it go back into its place on the table?
- “The square: r and f”: it is worth pausing on the rf and fr challenges; it is the first time order matters.
- The 3D rooms work well projected; the view rotates by dragging and turns are made by tapping the ends of the axes.
- “The group zoo” is also useful as a reference during lab II.
Room-by-room contents
Room 1 · The phone in its case
A drawn phone with a front (screen, clock) and a back (cameras) that can be turned half a turn, flipped left to right or top to bottom, with animation. A gallery fills with the positions found without saying how many there are; at four, the actions e, r, h, v are named and combined. Designed for phones: the teacher can do it first with their own phone.
Student tasks
- Find the four positions.
- Decide which single action equals two or three in a row (five challenges).
Room 2 · The triangle in its hole
A triangle with numbered corners and two faces of different colours; actions turn 120° and flip. Students find the six positions and then take the triangle to goal positions in the fewest moves (taking longer sends it back to the start). It closes with the pentagon: 2n positions, the group Δₙ.
Student tasks
- Find the six positions.
- Reach three goals in the fewest moves.
- Count the positions of the pentagon.
Room 3 · The square: r and f
The 8 positions of the square and their names rᵏ and rᵏf, read from left to right. Challenges of “where does it end up after rf, fr, r²f…?” with icons of each position. rf and fr differ: the first non-abelian group.
Student tasks
- Count the positions of the square.
- Identify the position of five words.
Room 4 · The four rules
Carter’s four rules (fixed actions that can be undone, are predictable and can be chained) and six examples to decide whether they form a group and which rule fails: random turn, clock, painting, perfect shuffle, adding stones, Rubik’s cube.
Student tasks
- Decide in six examples whether there is a group and, if not, which rule fails.
Room 5 · The map of the phone
As the phone moves, its Cayley diagram draws itself: one node per position (with its icon) and a line in the colour of each action. Then words are followed on the map.
Student tasks
- Complete the map.
- Follow four words and say where they end.
Room 6 · The map of the triangle
The diagram of Δ₃ with two concentric triangles: the inner one runs r backwards. Students tap the node each word reaches; rf and fr end in different places.
Student tasks
- Follow four words.
- Compare rf and fr.
- Explain why the inner triangle runs backwards.
Room 7 · Two paths, one place
Two paths to the same node are an equation: r³ = e, f² = e, fr = r²f. Challenges of writing a shorter word equal to a given one (Δ₃ and Δ₄), with an on-screen keyboard; every valid answer is accepted and both paths are animated.
Student tasks
- Shorten six words.
Room 8 · Other generators, another picture
Δ₃ with two flips gives a hexagon; Z₆ with r² and r³ gives a prism. Same group, different picture. Which elements generate Z₆.
Student tasks
- Recognise the same group in two diagrams.
- Decide which powers of r generate Z₆.
Room 9 · Impossible diagrams
Five diagrams: one is missing an arrow, in another two arrows of the same colour leave or enter a node, another is not regular and one is correct. Students say what is wrong.
Student tasks
- Diagnose five diagrams.
Room 10 · The table comes from the map
Cell (a, b) = go to a and follow the path of b, animated on the diagram. Cells of Z₄ and Δ₃ are filled in; (r, f) and (f, r) come out different.
Student tasks
- Fill in six cells.
Room 11 · A very special sudoku
Each row and column contains each element once, because everything can be undone. Complete tables of Z₄, Z₂×Z₂ and Z₅ using only that rule, and recognise tables that cannot belong to a group (repeated row, no identity, non-associative with its concrete triple).
Student tasks
- Complete three tables.
- Find the impossible table in three sets.
Room 12 · Reading a table of colours
Colour-only tables: abelian ⇔ symmetric about the diagonal; find the identity and inverses.
Student tasks
- Decide whether three tables are abelian.
- Find the identity and two inverses.
Room 13 · Repeat until you are back
The order of an element on the Z₁₂ clock (a token runs through its powers) and on the square.
Student tasks
- Compute seven orders.
Room 14 · Cycle graphs
Each element and its powers form a cycle through e. Count elements of each order and recognise the group of a graph.
Student tasks
- Count elements of order 2 and 4 in three groups.
- Identify two graphs.
Room 15 · Permutations
Cable diagrams and cycle notation (any equivalent way of writing it is accepted). Compose permutations and see that Δ₃ is the permutations of the corners: Δ₃ = S₃.
Student tasks
- Write two shuffles as cycles.
- Compose two pairs of permutations.
- Identify r and f as permutations.
Room 16 · Multiplying permutations
Each number is followed through two stacked cable diagrams: first σ, then τ, reading from left to right (with a warning that many books write it the other way round). A guided challenge number by number, then products in S₃ and S₄ in cycle notation, στ versus τσ, squares and inverses.
Student tasks
- Compute eight products, squares and inverses of permutations.
Room 17 · The cycle type decides
The order of a permutation is the lcm of its cycle lengths, checked by repeating it. Powers and inverses of cycles and counts by type in S₄ and S₅.
Student tasks
- Compute orders, powers and inverses.
- Count permutations of a given type in S₄.
Room 18 · Even and odd
Two definitions of parity that agree: the crossings in the cable diagram (visual) and transpositions, k − 1 per cycle (classical); Aₙ is the even ones.
Student tasks
- Decide the parity of four permutations.
- Give |A₄| and |A₅|.
Room 19 · The same group?
Animation of Z₂×Z₃ turning into Z₆. Five isomorphism questions decided by invariants (order, abelian, distribution of orders).
Student tasks
- Decide five isomorphisms.
Room 20 · Renaming without breaking anything
f(ab) = f(a)f(b) with two tables side by side, Z₄ and Z₅*: choose f(r), see the dictionary and check cell by cell (green or red). The Z₅* table is painted with Z₄’s colours through f. Also Δ₃ → S₃ and the phone versus Z₈*.
Student tasks
- Build an isomorphism Z₄ → Z₅*.
- Find the cell where a bijection that is not a homomorphism fails.
Room 21 · Z₈ is not Z₄×Z₂
Three animations: the race of orders, the attempted isomorphism that fails for every possible image of r, and the cycle graphs side by side. Z₂×Z₂×Z₂ as a third abelian group of order 8.
Student tasks
- Solve seven challenges on orders and isomorphism.
Room 22 · Multiplying modulo n
The invertible residues modulo n form the group Z*ₙ: a multiplication clock and a table. Orders, primitive roots and identification: Z₅* ≅ Z₄, Z₈* ≅ Z₂×Z₂, Z₇* ≅ Z₆, Z₁₅* ≅ Z₄×Z₂.
Student tasks
- Find the invertible residues.
- Compute orders.
- Identify four groups Z*ₙ.
Room 23 · Cayley’s theorem
Each element g moves the nodes of its own diagram (x ↦ xg): those arrows are a permutation, and every group is a subgroup of Sₙ. Animated example with Z₃ ⊂ S₃; permutations of Z₄ and V₄; then searching S₄ (a palette of its 24 elements) for subgroups isomorphic to Z₄, Z₂, V₄ and Δ₄, and seeing both diagrams side by side with the same positions and colours.
Student tasks
- Read three permutations of the regular representation.
- Find subgroups of S₄ isomorphic to Z₄, Z₂, V₄ and Δ₄.
Room 24 · The turns of the tetrahedron
A 3D tetrahedron with axes you tap to turn. Count axes and rotations: 12 ≅ A₄.
Student tasks
- Count axes and turns by type.
- Compose two turns.
- Recognise A₄.
Room 25 · The turns of the cube
The 24 rotations by type of axis (faces, diagonals, edges). With the four diagonals painted, it turns out to be S₄.
Student tasks
- Count the turns of each type.
- Recognise S₄.
Room 26 · The turns of the icosahedron
60 rotations: 24 about vertices, 20 about faces, 15 about edges and e ≅ A₅. Each challenge shows only the axes it asks about.
Student tasks
- Count axes and turns by type.
- Recognise A₅.
Room 27 · Cayley diagrams in 3D
Cayley diagrams as the orbit of a point: A₄ is a truncated tetrahedron, S₄ a rhombicuboctahedron, A₅ a truncated icosahedron. Read the orders of the generators off their faces.
Student tasks
- Count nodes.
- Read the orders of a and ab.
- Recognise the solid of S₄.
Room 28 · The 2×2×2 Rubik’s cube
A 3D cube with animated turns and a button to repeat a formula while counting. The order of R, R U, R U', R U2, R U R' U' and R2 U2.
Student tasks
- Find the order of six formulas.
Room 29 · The group zoo
An explorer with the groups of up to 12 elements, S₄ and A₅: table, cycle graph, Cayley diagram and properties. Six mystery groups by their properties.
Student tasks
- Identify six groups by their properties.
Rooms to project
The most striking ones to show and discuss in class.