# Exceptional character

In mathematical finite group theory, an **exceptional character** of a group is a character related in a certain way to a character of a subgroup. They were introduced by Suzuki (1955, p. 663), based on ideas due to Brauer in (Brauer & Nesbitt 1941).

Suppose that *H* is a subgroup of a finite group *G*, and *C*_{1}, ..., *C*_{r} are some conjugacy classes of *H*, and φ_{1}, ..., φ_{s} are some irreducible characters of *H*.
Suppose also that they satisfy the following conditions:

Then *G* has *s* irreducible characters *s*_{1},...,*s*_{s}, called **exceptional characters**, such that the induced characters φ_{i}* are given by

where ε is 1 or −1, *a* is an integer with *a* ≥ 0, *a* + ε ≥ 0, and Δ is a character of *G* not containing any character *s*_{i}.

The conditions on *H* and *C*_{1},...,*C*_{r} imply that induction is an isometry from generalized characters of *H* with support on *C*_{1},...,*C*_{r} to generalized characters of *G*. In particular if *i*≠*j* then (φ_{i} − φ_{j})* has norm 2, so is the difference of two characters of *G*, which are the exceptional characters corresponding to φ_{i} and φ_{j}.