Linear indpendence of characters

Posted on May 18, 2019

Theorem. Let G be an abelian group, let K be a field, and suppose that \chi_1,\ldots,\chi_n are distinct characters of G with values in K, i.e., \chi_i:G\to K^* is group homomorphism for each i=1,\ldots,n. Then the characters are linearly independent, i.e., if c_1,\ldots,c_n\in K such that      c_1\chi_1(g)+\ldots+c_n\chi_n(g)=0 for all g\in G, then it must be that c_1=\ldots c_n=0.

If n=1 observe that since \chi_1(g)\in K^* in particular \chi_1(g)\ne 0 for all g\in G. Thus c_1\chi_1(g)=0 implies c_1=0.

Suppose that for some n\ge 2 all sets of fewer than n characters of G are linearly indpendent. Let \chi_1,\ldots,\chi_{n} be distinct characters and c_1,\ldots,c_{n}\in K such that      c_1\chi_1(g)+\ldots+c_{n}\chi_{n}(g)=0 for all g\in G. Let i,j\in\{1,\ldots,n\} be distinct. Observe that since the characters are distinct there exists g_{ij}\in G such that \chi_i(g_{ij})\ne\chi_j(g_{ij}). Observe that by our choice of c_1,\ldots,c_n we have  \begin{aligned}[t]     0 &= c_1\chi(g_{ij}g)+\ldots+c_n\chi_n(g_{ij}g)\\       &= c_1\chi_1(g_{ij})\chi_1(g)+\ldots+c_n\chi_n(g_{ij})\chi_n(g)\\       &= \sum_{r=1}^n c_r\chi_r(g_{ij})\chi_r(g) \end{aligned} and  \begin{aligned}[t]     0 &=\chi_j(g_{ij})(c_1\chi(g)+\ldots+c_n\chi_n(g))\\       &=c_1\chi_j(g_{ij})\chi_1(g)+\ldots+c_n\chi_j(g_{ij})\chi_n(g)\\       &= \sum_{r=1}^n c_r\chi_j(g_{ij})\chi_r(g) \end{aligned} for all g\in G. Therefore subtracting the two we see that      \sum_{r=1}^{n}c_r(\chi_r(g_{ij})-\chi_j(g_{ij}))\chi_r(g)=0 for all g\in G. Since the term r=j vanishes, by the inductive hypothesis we must have all the coefficients of the linear combination equal to zero. That is, we must have c_r(\chi_r(g_{ij})-\chi_j(g_{ij}))=0 for all r=1,\ldots,n. In particular, c_i(\chi_i(g_{ij})-\chi_j(g_{ij}))=0. But by our choice of g_{ij} we have \chi_{i}(g_{ij})-\chi_j(g_{ij})\ne 0. Thus c_i=0.

The above proof is inspired by the proof given in (1, Theorem 2.1).


  1. Keith Conrad, Linear independence of characters