Rational group

A finite group G is rational (a Q-group) if every value of every complex irreducible character of G is a rational number — equivalently a rational integer, so the entire character table is integer-valued.

Equivalent condition

For every g ∈ G:

• g is conjugate to gᵏ whenever gcd(k, ord(g)) = 1
• i.e. g is conjugate to every generator of ⟨g⟩

Facts

• The character table is integer-valued.
• Every symmetric group Sₙ is rational.
• Weyl groups of crystallographic root systems are rational.
• Q₈ and every elementary abelian 2-group are rational.

Examples

• S₃, S₄ — all symmetric groups are rational.
• Q₈.
• C₂×C₂ — every elementary abelian 2-group.

Non-example

C₃ is not rational: it is abelian, so a generator g is not conjugate to g² = g⁻¹. A nontrivial irreducible character therefore takes the value of a primitive cube root of unity ζ₃, which is irrational.

See also

• conjugacy
• symmetric
• abelian