The character table of a finite group G is always a square matrix. Why?
Think about your answer, then reveal below.
Model answer: The number of rows equals the number of non-isomorphic irreducible representations, and the number of columns equals the number of conjugacy classes. These two numbers are always equal for finite groups over ℂ, making the table square.
This equality is a deep fact. The irreducible characters form an orthonormal basis for the space of class functions on G, and the dimension of this space equals the number of conjugacy classes (since a class function is determined by its values on conjugacy classes). So the number of irreducible characters must equal the number of conjugacy classes.
Question 2 Multiple Choice
In the character table of S₃, the dimensions of the irreducible representations are 1, 1, and 2. What relation must these dimensions satisfy?
ATheir sum equals |S₃| = 6
BTheir product equals |S₃| = 6
CThe sum of their squares equals |S₃| = 6
DThe sum of their cubes equals |S₃| = 6
The sum-of-squares formula states that Σ dᵢ² = |G|, where dᵢ = χᵢ(e) is the dimension of the i-th irreducible representation. For S₃: 1² + 1² + 2² = 1 + 1 + 4 = 6 = |S₃|. This follows from decomposing the regular representation: it contains each irreducible Vᵢ with multiplicity dᵢ, so its dimension |G| = Σ dᵢ².
Question 3 True / False
Every entry in the character table of a finite group is an algebraic integer.
TTrue
FFalse
Answer: True
Each χᵢ(g) is a sum of eigenvalues of ρᵢ(g). Since g has finite order n, ρᵢ(g)ⁿ = I, so each eigenvalue is an nth root of unity — and roots of unity are algebraic integers. A sum of algebraic integers is an algebraic integer. This arithmetic constraint, combined with the orthogonality relations, is a powerful tool for computing character tables: entries must be algebraic integers that satisfy specific inner product equations.
Question 4 Multiple Choice
Can two non-isomorphic groups have identical character tables?
ANo — the character table determines the group up to isomorphism
BYes — for example, the dihedral group D₄ and the quaternion group Q₈ have the same character table
CYes — but only for abelian groups
DNo — because the orthogonality relations uniquely determine the table from the group
D₄ and Q₈ are non-isomorphic groups of order 8 (D₄ has elements of order 4 while Q₈ has a unique element of order 2), yet they have identical character tables — both have five conjugacy classes and irreducible representations of dimensions 1, 1, 1, 1, 2. This shows the character table does not determine the group: it captures the 'representation-theoretic' structure but loses some information about the group's element-level structure.