Questions: Representations of Abelian Groups

Question 1

How many irreducible complex representations does ℤ/2ℤ × ℤ/2ℤ have, and what are their dimensions?

• A. 2 representations, each of dimension 2
• B. 4 representations, each of dimension 1
• C. 1 representation of dimension 4
• D. 3 representations of dimensions 1, 1, 2

Answer: (B)

The group has order 4 and is abelian, so all irreducible representations are 1-dimensional. There must be exactly 4 of them (one per element, since each element is its own conjugacy class). They are: (1,1)→1, (1,−1), (−1,1), (−1,−1), where each entry gives the scalar by which the corresponding generator acts. Check: 1² × 4 = 4 = |G|.

Question 2

The dual group Ĝ = Hom(G, ℂ*) of a finite abelian group G is isomorphic to G.

• T. True
• F. False

Answer: True

For G = ℤ/nℤ, Ĝ ≅ ℤ/nℤ (there are n homomorphisms to ℂ*, indexed by roots of unity). For G = ℤ/n₁ℤ × ··· × ℤ/nₖℤ, Ĝ ≅ ℤ/n₁ℤ × ··· × ℤ/nₖℤ ≅ G. The isomorphism is non-canonical — it depends on choosing generators — but the abstract group structure is the same. This is a special case of Pontryagin duality for finite abelian groups.

Question 3

Let G = ℤ/2ℤ × ℤ/3ℤ. The representation sending (a,b) to (−1)ᵃ · ω^b (where ω = e^{2πi/3}) is irreducible because:

Model answer: It is 1-dimensional (a homomorphism G → ℂ*), and all 1-dimensional representations are automatically irreducible since they have no proper nonzero subspaces.

This representation maps (1,0) ↦ −1 and (0,1) ↦ ω. Since (−1)² = 1 and ω³ = 1, it is a well-defined group homomorphism to ℂ*. It is one of the 6 = 2·3 irreducible representations of ℤ/2ℤ × ℤ/3ℤ, obtained by choosing one representation from each cyclic factor independently.

Question 4

For a finite abelian group G, the character table is a |G| × |G| matrix. What special property does this matrix have?

• A. It is the identity matrix
• B. All entries are ±1
• C. Its rows are orthogonal with respect to the standard inner product (after conjugation), and it defines a generalized DFT on G
• D. It is upper triangular

Answer: (C)

The character table of a finite abelian group has rows indexed by Ĝ and columns by G. The orthogonality relations say the rows are orthogonal: Σ_{g∈G} χ(g)·conjugate(χ'(g)) = |G|·δ_{χ,χ'}. The matrix (1/√|G|)(χ(g))_{χ,g} is unitary and defines the Fourier transform on G. For cyclic groups this recovers the DFT; for general abelian groups it is the higher-dimensional DFT on the product structure.