# the sum of the values of a character of a finite group is $0$

The following is an argument that occurs in many proofs involving characters of groups. Here we use additive notation for the group $G$, however this group is not assumed to be abelian^{}.

###### Lemma 1.

Let $G$ be a finite group^{}, and let $K$ be a field. Let $\chi \mathrm{:}G\mathrm{\to}{K}^{\mathrm{\times}}$ be a character, where ${K}^{\mathrm{\times}}$ denotes the multiplicative group^{} of $K$. Then:

$$\sum _{g\in G}\chi (g)=\{\begin{array}{cc}\mid G\mid ,\mathit{\text{if}}\chi \mathit{\text{is trivial,}}\hfill & \\ {0}_{K},\mathit{\text{otherwise}}\hfill & \end{array}$$ |

where ${\mathrm{0}}_{K}$ is the zero element in $K$, and $\mathrm{\mid}G\mathrm{\mid}$ is the order of the group $G$.

###### Proof.

First assume that $\chi $ is trivial, i.e. for all $g\in G$ we have $\chi (g)=1\in K$. Then the result is clear.

Thus, let us assume that there exists ${g}_{1}$ in $G$ such that $\chi ({g}_{1})=h\ne 1\in K$. Notice that for any element ${g}_{1}\in G$ the map:

$$G\to G,g\mapsto {g}_{1}+g$$ |

is clearly a bijection. Define $\mathcal{S}={\sum}_{g\in G}\chi (g)\in K$. Then:

$h\cdot \mathcal{S}$ | $=$ | $\chi ({g}_{1})\cdot \mathcal{S}$ | ||

$=$ | $\chi ({g}_{1})\cdot {\displaystyle \sum _{g\in G}}\chi (g)$ | |||

$=$ | $\sum _{g\in G}}\chi ({g}_{1})\cdot \chi (g)$ | |||

$=$ | $\sum _{g\in G}}\chi ({g}_{1}+g),(1)$ | |||

$=$ | $\sum _{j\in G}}\chi (j),(2)$ | |||

$=$ | $\mathcal{S}$ |

By the remark above, sums $(1)$ and $(2)$ are equal, since both run over all possible values of $\chi $ over elements of $G$. Thus, we have proved that:

$$h\cdot \mathcal{S}=\mathcal{S}$$ |

and $h\ne 1\in K$. Since $K$ is a field, it follows that $\mathcal{S}=0\in K$, as desired.

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Title | the sum of the values of a character of a finite group is $0$ |
---|---|

Canonical name | TheSumOfTheValuesOfACharacterOfAFiniteGroupIs0 |

Date of creation | 2013-03-22 14:10:30 |

Last modified on | 2013-03-22 14:10:30 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 6 |

Author | alozano (2414) |

Entry type | Theorem |

Classification | msc 11A25 |