In representation theory, a branch of mathematics, Artin's theorem, introduced by E. Artin, states that a character on a finite group is a rational linear combination of characters induced from all cyclic subgroups of the group.
There is a similar but somehow more precise theorem due to Brauer, which says that the theorem remains true if "rational" and "cyclic subgroup" are replaced with "integer" and "elementary subgroup".
In Linear Representation of Finite Groups Serre states in Chapter 9.2, 17 [1] the theorem in the following, more general way: Let
finite group,
Then the following are equivalent: This in turn implies the general statement, by choosing
be a finite groupe and
Let us denote, like Serre did in his book,
's characters are a linear combination of
with positive integer coefficient, the elements of
forms a basis ), which, by tensor product, is isomorphic to
to one of its subgroup and its dual operator
of induction of a representation can be extended to an homomorphisme :
With those notations, the theorem can be equivalently re-write as follow : If
, the following properties are equivalents : This result from the fact that
is of finite type.
Before getting to the proof of it, understand that the morphisme
Let’s begin the proof with the implication 2.
Starting with the following lemma : Let
It is enough to prove this assertion for the character
,by definition, V is the direct sum of the transformed
In such a basis, the diagonal of the matrix of
( which is ruled out by hypothesis ), it is fully null, we thus have
which conclude the proof of the lemma.
, which in turn means the multiples of
are distinct elements of the cokernel of
composed of element of the image
), which, through duality, is equivalent to prove the injectivity of
Which is obvious : indeed this is equivalent to say that if a class function is null on ( at least ) one element of each class of conjugation of
, it is null ( but class function are constant on conjugation class ).
This conclude the proof of the theorem.