Orbit sizes for irreducible representations may differ from orbit sizes for conjugacy classes under action of automorphism group

Statement

It is possible to have a finite group $G$ with the property that under the action of the automorphism group $\operatorname {Aut} (G)$, the orbit sizes for the set $R(G)$ of irreducible linear representations of $G$ are not the same as the orbit sizes for the set $C(G)$ of conjugacy classes of $G$.

A finite group where the orbit sizes are in fact the same is termed a finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group.

Related facts

Opposite facts

The most direct opposite fact is: Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group‎

This gives a sufficient condition for being a finite group having the same orbit sizes of conjugacy classes and irreducible representations under automorphism group. The following hold for all finite groups:

• Number of irreducible representations equals number of conjugacy classes
• Number of orbits of irreducible representations equals number of orbits under automorphism group
• Number of irreducible representations over reals equals number of equivalence classes under real conjugacy
• Number of irreducible representations over rationals equals number of equivalence classes under rational conjugacy
• Number of irreducible representations over complex numbers with real character values equals number of conjugacy classes of real elements

Similar facts

• Number of irreducible representations over rationals need not equal number of conjugacy classes of rational elements

Proof

There are examples of groups of order 27.

External links

• This MathOverflow discussion