By Aschbacher M.
Finite crew conception is meant to serve either as a textual content and as a easy reference on finite teams. In neither function do I want the booklet to be encyclopedic, so i have incorporated in basic terms the fabric I regard as so much basic. whereas such judgments are subjective, i have been guided by way of a number of easy rules which i believe are vital and may be made specific.
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Example text
R}. 2. Proof. Set K = N G. Indeed, since N r k=1 ker χk = {e}. r k=1 ker χk G. 1, for each k, ker χk = ker ρk , hence ker ρk there is a factorisation of ρk : G −→ GLC (Vk ), p ρ k G→ − G/K −→ GLC (Vk ), where p : G −→ G/K is the quotient homomorphism. As p is surjective, it is easy to check that ρk is an irreducible representation of G/K, with character χk . Clearly the χk are distinct irreducible characters of G/K and nk = χk (e) = χk (eK) are the dimensions of the corresponding irreducible representations.
Characters and the structure of groups In this section we will give some results relating the character table of a finite group to its subgroup structure. Let ρ : G −→ GLC (V ) be a representation for which dimC V = n. 1. ker χρ = ker ρ and hence ker χρ is a normal subgroup of G. Proof. For g ∈ ker χρ , let v = {v1 , . . , vn } be a basis of V consisting of eigenvectors of ρg , so ρg vk = λk vk for suitable λk ∈ C, and indeed each λk is a root of unity and so has the form λk = etk i for tk ∈ R.
S = k=1 But then λj = (ψs |χj ). We also have (ψs |χj ) = 1 |G| r = k=1 = ψs (g)χj (g) g∈G ψs (gk )χj (gk ) | CG (gk )| χj (gs ) , | CG (gs )| hence r ψs = j=1 χj (gs ) χj . | CG (gs )| Thus we have the required formula r δst = ψs (gt ) = j=1 χj (gt )χj (gs ) . | CG (gs )| 5. Examples of character tables Equipped with the results of the last section, we can proceed to find some character tables. For abelian groups we have the following result which follows from what we have seen already together with the fact that in an abelian group every conjugacy class has exactly one element.