Linear indpendence of characters

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).

References

Keith Conrad, Linear independence of characters