Teaching objectives
Third group theory lab, for first-year university. It follows “Cosets and quotients” and studies maps between groups that respect the structure: what is lost, what is kept, and when two groups are really the same.
The thread is that a homomorphism is a map that respects arrows: each element of G lights up, an arrow carries it to its image, and the powers of a generator go to the powers of its image. What the map crushes is the kernel, what it reaches is the image, and the first isomorphism theorem ties them to the quotient of the previous lab. From there students recognise isomorphic groups, split groups into products and count automorphisms, up to the cube, which gives a homomorphism from S₄ to S₃, and a last room with a zoo of quotients G/N ≅ Q. It is followed by “Centre, commutators and solvable groups”.
What students learn
- To understand homomorphisms as maps that respect arrows, and to use a test bench to see when an assignment of generators defines a homomorphism and when it leads to a contradiction.
- To compute the kernel and the image, and to use the formula |G| = |kernel| · |image|.
- To state and see the first isomorphism theorem: G/kernel ≅ image.
- To see that the order of φ(g) divides that of g, and that it is preserved when φ is injective (in particular, by an isomorphism); to compute orders in Z₁₃* and Z₂₁* and use them to count homomorphisms and rule out isomorphisms.
- To pair up isomorphic groups (V₄ and Z₂×Z₂, Z₂×Z₃ and Z₆, Δ₃ and S₃…) and spot those with no partner.
- To recognise direct products in diagrams and decide when Zₘ×Zₙ is cyclic: exactly when gcd(m, n) = 1.
- To decide whether a group is a product by looking at invariants: the distribution of orders and whether it is abelian.
- To compute the automorphisms of Zₙ (r ↦ rᵏ with gcd(k, n) = 1) and see that Aut(Zₙ) ≅ Uₙ.
- To count the automorphisms of Z₂×Z₂ and other small groups, and the homomorphisms from S₄ to S₃.
Key mathematical ideas
- A homomorphism is decided on the generators. It is enough to say where they go; if the relations are not respected, a contradiction appears.
- The kernel is always normal, and the image is a subgroup of the target. In the lab the kernel is painted violet and the image amber.
- The first isomorphism theorem. Collapsing the table by the kernel gives exactly the table of the image, with the same colours.
- ord φ(g) divides ord g. Because φ(g)ⁿ = φ(gⁿ) = e. If φ is injective they are equal, so isomorphic groups have the same distribution of orders.
- Zₘ×Zₙ ≅ Zₘₙ if gcd(m, n) = 1. Otherwise the product has no element of order mn.
- Aut(Z₈) ≅ Z₂×Z₂ and Aut(Z₂×Z₂) ≅ S₃. Automorphisms form a group with a structure of its own.
- There are 10 homomorphisms from S₄ to S₃. Six are onto (with kernel V₄, of 4 elements), three have an image of 2 elements, and one is trivial.
- The rotations of the cube rearrange its three axes. This gives a homomorphism from S₄ to S₃ with a kernel of 4 elements (Arnold).
- A quotient is recognised in its table. Collapsing the table of G by the cosets of N gives the table of a known group: Δ₈/Z(Δ₈) ≅ Δ₄, Q₈/Z(Q₈) ≅ Z₂×Z₂, A₄/V₄ ≅ Z₃… The isomorphism says which coset goes to which element, and it is not unique.
Sources and credits
The pictures, the homomorphism test bench and the isomorphisms follow Visual Group Theory by Nathan Carter (MAA, 2009). The cube homomorphism follows Abel’s Theorem in Problems and Solutions by V. B. Alekseev, based on V. I. Arnold’s lectures.
Classroom notes
- “Maps that respect arrows”: let students try absurd assignments; the red contradiction (for example Z₃ → Z₂ with r ↦ r) teaches more than the correct one.
- “Kernel and image” and “The first isomorphism theorem” are best done back to back, with the table by blocks in view.
- “Isomorphic pairs” has traps: groups with no partner. Ask students to justify with an invariant.
- “The cube and its three axes” can be projected with the cube turning while counting the axes aloud.
- “A zoo of quotients” closes the lab: project Δ₈/Z(Δ₈) ≅ Δ₄ and, before collapsing, ask which group of 8 elements it will look like; then Z₄×Z₂ modulo two different subgroups of 2 elements gives two different quotients.
Room-by-room contents
Room 1 · Maps that respect arrows
Animation: each element of G lights up and an arrow takes it to its image in H; the powers of a generator go to the same powers of its image. Three demos: Z₆ → Z₃ (r ↦ r), Z₃ → Z₂ (r ↦ r, contradiction on returning to e) and Z₃ → Δ₃ (the image is only the 3 rotations). In each challenge, a test bench: assign the image of each generator and “Follow the arrows” propagates and marks the contradiction if there is one.
Student tasks
- Build homomorphisms Z₄ → Z₂, the two Δ₄ → Z₂ that send one generator to e and the other to r, an injective Z₄ → Z₈ and a non-trivial Z₃ → Δ₃.
- Decide whether there is a non-trivial Z₅ → Z₃ (no), count the non-trivial Z₆ → Z₄ (1) and whether there is an injective Z₆ → Δ₃ (no: Z₆ is abelian and Δ₃ is not).
- Where e always goes.
Room 2 · Kernel and image
Animated demo Z₆ → Δ₃ (r ↦ r): kernel {e, r³}, image the 3 rotations, 6 = 2 · 3. |G| = |kernel| · |image| with Δ₄ → Z₂, Z₁₂ → Z₄, Z₆ → Z₆, Z₆ → Δ₃ (r ↦ s, image {e, s}), Z₄ → Z₈ (injective, image 4 of 8) and the sign of S₄.
Student tasks
- Compute kernels and images, also when the image is not the whole group.
- Recognise A₄ as the kernel of the sign.
Room 3 · The first isomorphism theorem
G/ker φ ≅ im φ seen with the table: G is sorted by cosets of the kernel, collapsed, and the table of the image appears beside it in the same colours (gN ↦ φ(g)). Demos: Δ₄ → Z₂ (rotations to e, flips to r) and Z₆ → Δ₃ (r ↦ r, the image is not all of Δ₃).
Student tasks
- Recognise G/kernel for Z₁₂ → Z₄, Z₁₂ → Z₆, Δ₆ → Δ₃ and Q₈ → Z₂×Z₂.
- Count the image of A₄ → Z₃ (3).
Room 4 · Order and homomorphisms
Theorem with formulas: if g has order n, φ(g)ⁿ = φ(gⁿ) = φ(e) = e, so ord φ(g) divides ord g; if φ is injective they are equal, and an isomorphism preserves the distribution of orders. Visual: two wheels of powers turning together (g on the left, φ(g) on the right); each gᵏ is painted with the colour of its image. Demos Z₁₂ → Z₄ (the wheel of 4 makes 3 turns) and Z₄ → Z₈ with r ↦ r² (injective: same order).
Student tasks
- Orders in Z₁₃*: of 2 (12) and of 3 (3); how many elements of order 12 (4).
- Where an injective homomorphism Z₄ → Z₁₃* sends r (5 or 8) and how many homomorphisms Z₆ → Z₁₃* there are (6).
- Elements of order 2 in Z₂₁* (3) and whether there is an injective homomorphism Z₁₂ → Z₂₁* (no).
Room 5 · Isomorphic pairs
Isomorphic = the same table up to names: V₄ and Z₂×Z₂ in the same colours; Z₄ is not (it has an element of order 4). Four rounds of matching group cards, with decoys that have no pair: Zₘ×Zₙ and Zₘₙ, units (Z₁₃*, Z₂₁*, Z₁₆*, Z₂₄*, with the orders from the previous room), Δ₃ and S₃, Δ₆ and Δ₃×Z₂, the cube and S₄, Q₈ and Δ₄… After the first round, the isomorphism Z₁₂ → Z₃×Z₄ is built on the arrow test bench by choosing the image of r.
Student tasks
- Match isomorphic groups and flag those with no pair, in four rounds.
- Send r to an element of order 12 of Z₃×Z₄ and check with the arrows that it is a bijection.
Room 6 · Inflating a diagram
The diagram of Zₘ×Zₙ is obtained by inflating one into the other. Zₘ×Zₙ is cyclic if and only if gcd(m, n) = 1.
Student tasks
- Decide whether six products are cyclic.
- State the gcd rule.
Room 7 · Is it a product?
Recognise products by invariants: Δ₆ ≅ Δ₃×Z₂, Z₆×Z₂ ≇ Z₁₂, Q₈ ≇ Z₄×Z₂…
Student tasks
- Decide six decompositions.
Room 8 · Automorphisms of Zₙ
An automorphism is an isomorphism from a group to itself. In Zₙ, r ↦ rᵏ is one exactly when gcd(k, n) = 1. Demos in Z₆: r ↦ r⁵ yes, r ↦ r² no (image {e, r², r⁴}). Aut(Z₈) with its composition table: multiply exponents mod 8.
Student tasks
- Find the 4 automorphisms of Z₅.
- Count those of Z₈ (4) and Z₉ (6).
- Recognise Aut(Z₈) ≅ Z₂×Z₂.
Room 9 · The automorphisms of Z₂×Z₂
With two generators a, b: every automorphism permutes the three elements of order 2. There are 6 and they form S₃. Contrast: Z₄, of the same size, has only 2.
Student tasks
- Find the 6 automorphisms of Z₂×Z₂.
- Recognise Aut(Z₂×Z₂) ≅ S₃.
- Count those of Z₄ (2).
Room 10 · Homomorphisms from S₄ to S₃
The test bench with S₄ drawn as 6 groups of 4 (the cosets of V₄): every homomorphism sends each group whole to one element of S₃. Demos: an onto one and one with a contradiction (c⁴ = e is not respected). There are 10: 6 onto, 3 with an image of 2 elements and the trivial one.
Student tasks
- Build an onto one and one with an image of 2 elements.
- Count them all (10).
- Give the kernel of an onto one (4).
Room 11 · The cube and its three axes
Second-to-last room. A 3D cube with its three face axes attached, each in its own colour up to the ends: each turn rearranges them in their 3 places. Animated examples: a quarter turn swaps two axes; a turn about a diagonal moves them in a cycle. The cube does not spin on its own, and the 3 places are fixed directions (left–right, top–bottom, front–back). Two tables compare the start with now: which axis lies in each direction (6 possibilities) and which faces are on top and in front (24, one per rotation). A homomorphism S₄ → S₃ with a kernel of 4 (Arnold).
Student tasks
- Find the kernel.
- Count the image.
- Recognise the quotient S₃.
Room 12 · A zoo of quotients
Last room. Ten showcases with pairs G/N ≅ Q: Δ₈/Z(Δ₈) ≅ Δ₄, Δ₄/Z(Δ₄) ≅ Z₂×Z₂, Q₈/Z(Q₈) ≅ Z₂×Z₂, Δ₆/⟨r²⟩ ≅ Z₂×Z₂, Z₈/⟨r⁴⟩ ≅ Z₄, Z₄×Z₂ modulo ⟨(0,1)⟩ (Z₄) and modulo ⟨(2,0)⟩ (Z₂×Z₂), Z₃×Z₃/⟨(1,1)⟩ ≅ Z₃, A₄/V₄ ≅ Z₃ and S₄/A₄ ≅ Z₂. In each one, the table of G by the cosets of N (one colour per coset) collapses into that of G/N and expands back; when collapsed, the table of Q appears next to it with the same colours, and below it the isomorphism: which coset goes to which element. Close of the lab.
Student tasks
- Build, coset by coset, an isomorphism Z₁₂/⟨r³⟩ → Z₃, Z₄×Z₂/⟨(2,1)⟩ → Z₄, Δ₆/Z(Δ₆) → Δ₃ and S₄/V₄ → S₃ (any one is accepted).
Rooms to project
The most striking ones to show and discuss in class.