Artin's induction theorem states that every complex group character 
of a finite group 
is a linear combination
with rational number coefficients of group
characters of induced representations
from subgroups that are cyclic
groups (tom Dieck 2009, Theorem 4.5.2). Thus
 |
where 
,
each 
is a cyclic subgroup of 
, and each 
is a one-dimensional group
character of 
. Multiplying 
by the group order 
gives a linear combination
of such induced group characters with integer
coefficients. The coefficients may be negative.
For example, let 
be the symmetric group,
let 
be generated by a transposition, and let 
be generated by a permutation
cycle of length 3. The trivial representation
has group character
 |
The identity can be checked on the three conjugacy classes of 
,
represented by the identity element, a transposition,
and a permutation cycle of length 3.
