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 CGmodule
(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 Pelementary subgroups.