If a subgroup 
of 
has a group representation 
, then there is a unique
induced representation of 
on a vector space 
. The original space 
is contained in 
, and in fact,
 |
(1)
|
where 
is a copy of 
.
The induced representation on 
is denoted 
.
Alternatively, the induced representation is the CG-module
 |
(2)
|
Also, it can be viewed as 
-valued functions on 
which commute with the 
action.
 |
(3)
|
The induced representation is also determined by its universal property:
 |
(4)
|
where 
is any representation of 
. Also, the induced representation satisfies the following
formulas.
1. 
.
2. 
for any group representation 
.
3. 
when 
.
Some of the group characters of 
can be calculated from the group
characters of 
, as induced representations, using Frobenius reciprocity.
Artin's reciprocity theorem says that
the induced representations of cyclic subgroups of
a finite group 
generates a lattice of finite
index in the lattice of virtual characters. Brauer's
theorem says that the virtual characters are generated by the induced representations
from P-elementary subgroups.
