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Let  $  K $
be a [[Field|field]], let  $  V $
be a finite-dimensional [[Vector space|vector space]] over  $  K $
and let  $  G $
be a [[Finite group|finite group]]. A representation of  $  G $
over  $  V $
is a group [[Homomorphism|homomorphism]]  $  X : G \rightarrow { { \mathop{\rm GL} } ( V/K ) } $(
the group of  $  K $-
linear automorphisms of  $  V $) 
or, equivalently, a module action of the [[Group algebra|group algebra]]  $  KG $
on  $  V $(
the equivalence is defined by:  $  g \cdot v = X ( g ) v $
for all  $  g,v \in V $; 
cf. also [[Representation of a group|Representation of a group]]). The character  $  {\chi _ {V} = { \mathop{\rm char} } ( X ) } : G \rightarrow K $
is defined by:  $  \chi _ {V} ( g ) = { \mathop{\rm tr} } ( X ( g ) ) $
for all  $  g \in G $. 
Since  $  { \mathop{\rm tr} } ( AB ) = { \mathop{\rm tr} } ( BA ) $
for any two  $  ( n \times n ) $-
matrices  $  A $, 
$  B $
over  $  K $, 
one finds that  $  \chi _ {V} ( ghg ^ {-1 } ) = \chi ( h ) $
for all  $  g,h \in G $, 
and hence  $  \chi _ {V} $
is a class function on  $  G: \chi _ {V} \in { \mathop{\rm CF} } ( G,K ) $. 
Equivalent representations have the same character and  $  \chi _ {V} $
is the sum of the characters of the quotient  $  KG $-
modules in any  $  KG $-
filtration of  $  V $. 
If  $  G $
acts irreducibly on  $  V/K $, 
then  $  V $
is said to be an irreducible  $  KG $-
module and  $  \chi _ {V} $
is said to be an irreducible character.

If  $  { \mathop{\rm dim} } ( V/K ) = 1 $, 
then  $  V $
is said to be a linear representation of  $  G $, 
and  $  V $
is an irreducible  $  KG $-
module;  $  \chi _ {V} $
is said to be a linear character. There are at most  $  | G | $
inequivalent types of irreducible representations of  $  G $
over  $  K $. 
Let  $  { \mathop{\rm Irr} } _ {K} ( G ) = \{ \chi _ {1} \dots \chi _ {k} \} $
be the set of irreducible characters of  $  G $
over  $  K $. 
Then  $  { \mathop{\rm Irr} } _ {K} ( G ) $
is  $  K $-
linearly independent in  $  { \mathop{\rm CF} } ( G,K ) $
and every character of  $  G $
over  $  K $
is a sum of elements of  $  { \mathop{\rm Irr} } _ {K} ( G ) $.

Let  $  H $
be a subgroup of  $  G $
and let  $  {\mathcal Y} : H \rightarrow { { \mathop{\rm GL} } ( W/K ) } $
be a representation of  $  H $. 
Clearly, the restriction to  $  H $, 
$  { { \mathop{\rm Res} } _ {H}  ^ {G} ( X ) } : H \rightarrow { { \mathop{\rm GL} } ( V/K ) } $, 
is a representation of  $  H $
and induction to  $  G $, 
$  { { \mathop{\rm Ind} } _ {H}  ^ {G} ( {\mathcal Y} ) } : G \rightarrow { { \mathop{\rm GL} } ( KG \otimes _ {KH }  W ) } $, 
is a representation of  $  G $(
cf. also [[Induced representation|Induced representation]]). If  $  T $
is a transversal for the right cosets of  $  H $
in  $  G $, 
then, for  $  g \in G $,

$$ 
{ \mathop{\rm char} } ( { \mathop{\rm Ind} } _ {H}  ^ {G} ( {\mathcal Y} ) ) ( g ) = \sum _ {t \in T } { \mathop{\rm char} } ( {\mathcal Y} )  ^ {o} ( tgt ^ {-1 } ) ,
$$

where  $  { \mathop{\rm char} }  ^ {o} ( {\mathcal Y} ) ( u ) = { \mathop{\rm Char} } ( {\mathcal Y} ) ( u ) $
if  $  u \in H $
and  $  { \mathop{\rm char} }  ^ {o} ( {\mathcal Y} ) ( u ) = 0 $
if  $  u \in G - H $.

If  $  V $
and  $  W $
are  $  KG $-
modules, then  $  V \otimes _ {K} W $
is a  $  KG $-
module defined by  "diagonal action" :

$$ 
g ( v \otimes _ {K} w ) \equiv ( gv ) \otimes _ {K} ( gw )
$$

for all  $  v \in V $, 
$  w \in W $
and  $  g \in G $, 
and  $  { \mathop{\rm char} } ( V \otimes _ {K} W ) = { \mathop{\rm char} } ( V ) { \mathop{\rm char} } ( W ) $.

Assume that  $  K = \mathbf C $, 
the field of complex numbers. Then  $  { \mathop{\rm Irr} } _ {\mathbf C} ( G ) $
is a  $  \mathbf C $-
basis of  $  { \mathop{\rm CF} } ( G, \mathbf C ) $. 
Also, every  $  \mathbf C G $-
module  $  V $
is completely reducible (i.e., a direct sum of irreducible submodules). Also,  $  \chi _ {V} ( 1 ) = { \mathop{\rm dim} } ( V/ \mathbf C ) $, 
$  \chi ( g ^ {- 1 } ) = {\overline{ {\chi ( g ) }}\; } $(
complex conjugate) and  $  \chi ( g ) $
is a sum of  $  \chi ( 1 ) $
$  | g | $-
th roots of unity for all  $  g \in G $. 
Also, there is a non-singular symmetric scalar product  $  {\langle  {\cdot, \cdot } \rangle } : { { \mathop{\rm CF} } ( G, \mathbf C ) \times { \mathop{\rm CF} } ( G, \mathbf C ) } \rightarrow \mathbf C $
defined by:

$$ 
\left \langle  {f _ {1} ,f _ {2} } \right \rangle = {
\frac{1}{\left | G \right | }
 } \sum _ {g \in G } f _ {1} ( g ) f _ {2} ( g ^ {-1 } ) .
$$

Here,  $  \langle  {\chi _ {i} , \chi _ {j} } \rangle = \delta _ {ij }  $(
the Kronecker delta) for all  $  \chi _ {i} , \chi _ {j} \in { \mathop{\rm Irr} } _ {\mathbf C} ( G ) $
and if  $  V $
and  $  W $
are two finite-dimensional  $  \mathbf C G $-
modules, then  $  { \mathop{\rm dim} } ( { \mathop{\rm Hom} } _ {\mathbf C G }  ( V,W ) / \mathbf C ) = \langle  {\chi _ {V} , \chi _ {W} } \rangle $
and hence the isomorphism type of  $  V $
is determined by  $  \chi _ {V} $.

If  $  p $
is a prime integer, then a [[Finite group|finite group]] that is the [[Direct product|direct product]] of a [[Cyclic group|cyclic group]] and a [[P-group| $  p $-
group]] (or equivalently of a cyclic  $  p  ^  \prime  $-
group and a  $  p $-
group) is called a  $  p $-
elementary group. Any subgroup or quotient of such a group is also  $  p $-
elementary. A finite group is called elementary if it is  $  p $-
elementary for some prime number  $  p $. 
It is well-known that each irreducible character  $  \chi $
of an elementary group  $  E $
is of the form  $  \chi = { \mathop{\rm Ind} } _ {H}  ^ {E} ( \psi ) $
for some subgroup  $  H $
of  $  E $
and some linear character  $  \psi $
of  $  H $(
cf. [[#References|[a8]]], Thm. 16).

For a finite group  $  G $, 
let  $  R ( G ) $
be the additive subgroup of  $  { \mathop{\rm CF} } ( G, \mathbf C ) $
generated by all characters of  $  G $. 
The elements of  $  R ( G ) $
are called virtual or generalized characters of  $  G $
and  $  R ( G ) $
is a ring and also a free Abelian group with free basis  $  { \mathop{\rm Irr} } _ {\mathbf C} ( G ) $. 
Clearly,

$$ 
{ \mathop{\rm Irr} } _ {\mathbf C} ( G ) = \left \{ {\varphi \in R ( G ) } : {\left \langle  {\varphi, \varphi } \right \rangle =1  \textrm{ and  }  \varphi ( 1 ) > 0 } \right \} .
$$

In [[#References|[a2]]], R. Brauer proved the following assertions:

1) Every character  $  \chi $
of a finite group  $  G $
is a linear combination with integer coefficients of characters induced from linear characters of elementary subgroups of  $  G $.

Brauer used this result in [[#References|[a2]]] to prove that Artin  $  L $-
functions of virtual characters have a meromorphic extension to the entire complex plane. Then, in [[#References|[a3]]], he proved that this assertion is equivalent to what is known as the Brauer characterization of characters:

2) A class function  $  \varphi \in { \mathop{\rm CF} } ( G, \mathbf C ) $
lies in  $  R ( G ) $
if and only if  $  { \mathop{\rm Res} } _ {E}  ^ {G} ( \varphi ) \in R ( E ) $
for every elementary subgroup  $  E $
of  $  G $.

An immediate consequence (cf. [[#References|[a8]]], Thm 22 and Corollary) is:

3) A class function  $  \varphi \in { \mathop{\rm CF} } ( G, \mathbf C ) $
lies in  $  R ( G ) $
if and only if for each elementary subgroup  $  E $
of  $  G $
and each linear character  $  \chi $
of  $  E $,

$$ 
\left \langle  { { \mathop{\rm Res} } _ {E}  ^ {G} ( \varphi ) , \chi } \right \rangle \in Z.
$$

A sort of converse of 1) was given by J. Green ([[#References|[a8]]], Thm. 23{'''}). There are numerous applications of these results (cf. [[#References|[a7]]], Lemma 8.14; Thm. 8.24, [[#References|[a6]]], V, Hauptsatz 19.11, [[#References|[a8]]], Sect. 11.2; Chap. 12).

Significant improvements to the proofs of these results have been obtained by several authors [[#References|[a4]]], [[#References|[a7]]], Chap. 8, [[#References|[a8]]], Chaps. 10, 11, [[#References|[a6]]], V, Sect. 19.

Let  $  R _ {+} ( G ) $
denote the free Abelian group whose free basis is given by the  $  G $-
conjugacy classes  $  ( H, \lambda ) $, 
where  $  H $
is a subgroup of  $  G $
and  $  \lambda $
is a linear character of  $  H $. 
Clearly  $  { \mathop{\rm Ind} } _ {H}  ^ {G} ( \lambda ) $
is a character of  $  G $
and hence induction induces an Abelian group homomorphism  $  {\mathcal I} : {R _ {+} ( G ) } \rightarrow {R ( G ) } $, 
which is surjective by 1). Some interesting recent results in [[#References|[a9]]] and [[#References|[a1]]] give explicit (functorial) splittings of  $  {\mathcal I} $(
i.e., an explicit group homomorphism  $  {\mathcal J} : {R ( G ) } \rightarrow {R _ {+} ( G ) } $
such that  $  {\mathcal I} \cdot {\mathcal J} = { \mathop{\rm id} } _ {R ( G ) }  $).

Clearly,  $  R ( G ) $
is the [[Grothendieck group|Grothendieck group]] of the category of finitely generated  $  \mathbf C G $-
modules. Consequently, 1) can be viewed as proving the surjectivity of the induction mapping from one Grothendieck group into another. By changing the coefficient ring  $  \mathbf C $, 
or by considering the modular context, etc., many important analogues of these results emerge, cf. [[#References|[a8]]], Chaps. 12, 16, 17, [[#References|[a5]]], Thm. 2, [[#References|[a10]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Boltje,   "A canonical Brauer induction formula"  ''Asterisque'' , '''181/2'''  (1990)  pp. 31–59</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Brauer,   "On Artin's <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110850/b110850162.png" />-series with general group characters"  ''Ann. of Math.'' , '''48'''  (1947)  pp. 502–514</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Brauer,   "A characterization of the characters of a group of finite order"  ''Ann. of Math.'' , '''57'''  (1953)  pp. 357–377</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R. Brauer,   J. Tate,   "On the characters of finite groups"  ''Ann. of Math.'' , '''62'''  (1955)  pp. 1–7</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  M. Broué,   "Sur l'induction des modules indecomposables et la projectraité relative"  ''Math. Z.'' , '''149'''  (1976)  pp. 227–245</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. Huppert,   "Endliche Gruppen" , '''I''' , Springer  (1967)  pp. Chapt. V</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  I.M. Isaacs,   "Character theory of finite groups" , Acad. Press  (1976)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.-P. Serre,   "Linear representations of finite groups" , Springer  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V. Snaith,   "Explicit Brauer induction"  ''Invent. Math.'' , '''94'''  (1988)  pp. 455–478</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  X. Zhou,   "On the decomposition map of Grothendieck groups"  ''Math. Z.'' , '''206'''  (1991)  pp. 533–534</TD></TR></table>