proof that dimension of complex irreducible representation divides order of group
Theorem Let be a finite group and an irreducible
complex representation of finite dimension
. Then divides.
Proof: Given any in the group ring of (denoted) we may define a sequence of submodules of (regarded as a module over )by equals the linear span of.
is Noetherian as a module over so wemust have for some . Hence may beexpressed as a linear combination of lower powers of. In other solves a monic polynomial ofdegree with coefficients in .
Given a conjugacy class in , we may set . Then is central in , as given , we have:
Hence applying to induces a linear map. By Schur’s lemma this must bemultiplication by somecomplex number . Then is an algebraicinteger
as it solves the same monic polynomial as .
Also any has finite order so the map it induces on must have eigenvalues which are roots of unity
and hence algebraicintegers. Hence the sum of the eigenvalues, , mustalso be an algebraic integer.
Now is irreducible so,
Therefore is both rational and an algebraic integer. Henceit is an integer and divides .