From: mareg@mimosa.csv.warwick.ac.uk () Subject: Re: One question of non-normal subgroup Date: 21 Dec 1999 13:31:03 GMT Newsgroups: sci.math Keywords: components of induced representations In article <83ne1p$bul$1@nnrp1.deja.com>, jrbao@my-deja.com writes: >There are very discussions of non-normal subgroup >in the group books. That's the reason reason I >have to post this question. I hope someone >can help me. I really need the answer. > >Suppose non-Abelian group G has a non-normal >subgroup K (K can be Abelian or non-Abelian). >Suppose Y is the representation of G induced by >trivial 1-dimensional representation of K. (As I >know, Y is called the principal induced >representation). Normally, Y is a reducible >representation of G, so we can suppose: >Y=1+X1+X2+...Xn. Here X1,X2, ... Xn are some >irreducible representations of G. My question is: >( I guess) At least one of X1, X2, ... Xn is > or >= 2 dimension. In other words, I want to know the >EXISTENCE of Xi, which is > or = 2 dimension. How >to proof it? Please help me. This is not too hard if you know a bit of character theory. Let \chi be the character of the induced representation Y. Then, using the standard product < , > defined on characters, < \chi, \chi > is equal to 1 + d_1^2 + ... + d_n^2, where d_i is the degree of the irreducible constituent X_i of Y. So, if all of the X_i had dimension 1, then we would have < \chi, \chi > = degree(\chi) = |G:K|, the index of K in G. On the other hand, by the Frobenius Reciprocity Theorem, < \chi, \chi > is equal to < 1_K, \chi_K > = (sum _ {k in K} \chi (k))/|K| (\chi is a real values cahracter), which by the (Burnside?) fixed point theorem is equal to the number of orbits of K in the permutation representation of G on the cosets of K. Now this number of orbits is equal to |G:K| if and only if K is a normal subgroup of G. Since you are assuming that K is not a normal subgroup, it follows that the X_i cannot all have dimension 1. Derek Holt.