Definition
Suppose that ''H'' is a subgroup of a finite group ''G'', and ''C''1, ..., ''C''''r'' are some conjugacy classes of ''H'', and φ1, ..., φ''s'' are some irreducible characters of ''H''. Suppose also that they satisfy the following conditions: #''s'' ≥ 2 #φ''i'' = φ''j'' outside the classes ''C''1, ..., ''C''''r'' #φ''i'' vanishes on any element of ''H'' that is conjugate in ''G'' but not in ''H'' to an element of one of the classes ''C''1, ..., ''C''''r'' #If elements of two classes are conjugate in ''G'' then they are conjugate in ''H'' #The centralizer in ''G'' of any element of one of the classes ''C''1,...,''C''''r'' is contained in ''H'' Then ''G'' has ''s'' irreducible characters ''s''1,...,''s''''s'', called exceptional characters, such that the induced characters φ''i''* are given by :φ''i''* = ε''s''''i'' + ''a''(''s''1 + ... + ''s''''s'') + Δ 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''.Construction
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''.See also
*References
* * *{{Citation , last1=Suzuki , first1=Michio , author1-link=Michio Suzuki (mathematician) , title=On finite groups with cyclic Sylow subgroups for all odd primes , jstor=2372591 , mr=0074411 , year=1955 , journal=