Let σ and τ be permutations of {1, 2, 3} where σ maps 1→2, 2→3, 3→1 and τ maps 1→2, 2→1, 3→3. What is the value of (σ∘τ)(1), where ∘ denotes composition (apply τ first, then σ)?
A1
B2
C3
DComposition is undefined because σ and τ conflict.
To compute (σ∘τ)(1): first apply τ to 1, giving τ(1) = 2; then apply σ to that result, giving σ(2) = 3. So (σ∘τ)(1) = 3. Notice that (τ∘σ)(1) = τ(σ(1)) = τ(2) = 1, which is different. This confirms that S₃ is non-abelian: σ∘τ ≠ τ∘σ. The order in which you apply permutations matters, just as it does for rotations in 3D space.
Question 2 Multiple Choice
Cayley's theorem states that every finite group is isomorphic to a subgroup of Sₙ for some n. What is the most significant practical and conceptual implication of this theorem?
AIt proves that all finite groups are abelian, since permutation groups are abelian.
BEvery abstract finite group can be concretely realized as a group of rearrangements of some set, so no finite group is truly 'exotic' — a permutation model always exists.
CIt shows that all groups must have order n! for some integer n.
DIt implies that studying Sₙ is sufficient to understand all of mathematics.
Cayley's theorem is an embedding result: it guarantees that even the most abstractly-defined finite group — specified only by its multiplication table — can be faithfully represented as permutations of some set. This provides a concrete computational handle on any abstract group. Note that S₃ is already non-abelian, so the theorem definitely does not imply all groups are abelian. The order of a group embedded in Sₙ divides n!, but need not equal n!.
Question 3 True / False
The symmetric group S₃ is non-abelian, meaning that the order in which two permutations are composed affects the result.
TTrue
FFalse
Answer: True
S₃ is the smallest non-abelian group. It has 3! = 6 elements, and there exist permutations σ, τ ∈ S₃ such that σ∘τ ≠ τ∘σ. For example, a cyclic rotation and a transposition in S₃ do not commute, as the computation above demonstrates. For n ≥ 3, Sₙ is always non-abelian. S₁ and S₂ are trivially abelian (they have only 1 and 2 elements, respectively).
Question 4 True / False
The symmetric group S₄ has 16 elements.
TTrue
FFalse
Answer: False
The order of Sₙ is n!, the number of ways to arrange n distinct objects. For S₄: |S₄| = 4! = 4 × 3 × 2 × 1 = 24, not 16. This is a common error — 16 = 2⁴ might arise from thinking each of 4 elements has 2 choices, but each element in a permutation is sent to a distinct image, so the count is n × (n-1) × (n-2) × ⋯ × 1. S₂ has 2 elements, S₃ has 6, S₄ has 24, S₅ has 120.
Question 5 Short Answer
Why does the order of composition matter in permutation groups (for n ≥ 3), and what algebraic property does this illustrate?
Think about your answer, then reveal below.
Model answer: Permutation composition is not commutative in general: applying permutation σ then τ can give a different result than applying τ then σ. This illustrates the non-abelian property: a group G is abelian if ab = ba for all elements, and non-abelian if this fails for some pair. In Sₙ (n ≥ 3), one can always find a cyclic rotation σ and a transposition τ such that σ∘τ ≠ τ∘σ. The result of sequentially rearranging a set depends on which rearrangement is done first — just as rotating then reflecting a physical object gives a different orientation than reflecting then rotating.
Non-abelian structure is one of the most important features of groups in algebra and physics. Sₙ is the prototype example — students first encounter non-commutativity here, which prepares them for matrix groups, Lie groups, and the symmetry groups of particles in physics. Cayley's theorem ensures this example is not exotic: it represents every finite group.