Ambivalent group
A finite group G is ambivalent (a real group) if every element g ∈ G is conjugate to its inverse g⁻¹.
Equivalent conditions
The following are equivalent:
- every element is conjugate to its inverse;
- every conjugacy class is closed under inversion (a real class);
- every complex irreducible character is real-valued — the whole character table is real.
Relation to rational groups
Every rational group is ambivalent — rational values lie in ℚ ⊂ ℝ — but not conversely: ambivalence only asks that character values be real, not rational.
Facts
- The character table is real-valued.
- An abelian group is ambivalent iff it has exponent ≤ 2 (every element is its own inverse) — i.e. it is an elementary abelian 2-group (or trivial).
- Every symmetric group Sₙ is ambivalent (indeed rational).
Q₈and every elementary abelian 2-group are ambivalent.
Examples
S₃,S₄— all symmetric groups.A₅— ambivalent but not rational (see below).Q₈.C₂×C₂— every elementary abelian 2-group.
Ambivalent but not rational
A₅ is ambivalent: each of its two classes of 5-cycles contains σ together with σ⁻¹. It is not rational, though — σ is not conjugate to σ², so a character separates them, taking the real but irrational values (1 ± √5) / 2. Real values that are not rational are exactly what distinguishes ambivalence from rationality.
Non-example
C₃ is not ambivalent: it is abelian of exponent 3, so a generator g is a distinct element from g⁻¹ = g² and cannot be conjugate to it. Its nontrivial irreducible character takes the value ζ₃, a non-real cube root of unity.