# proof that dimension of complex irreducible representation divides order of group

Theorem Let $G$ be a finite group^{} and $V$ an irreducible^{}
complex representation of finite dimension^{} $d$. Then $d$ divides
$|G|$.

Proof: Given any $\alpha $ in the group ring of $G$ (denoted $\mathbb{Z}G$) we may define a sequence of submodules of $\mathbb{Z}G$ (regarded as a module over $\mathbb{Z}$) by ${A}_{i}$ equals the $\mathbb{Z}$ linear span of $\{1,\alpha ,{\alpha}^{2},\mathrm{\cdots},{\alpha}^{i}\}$.

$\mathbb{Z}G$ is Noetherian as a module over $\mathbb{Z}$ so we
must have ${A}_{i}={A}_{i-1}$ for some $i$. Hence ${\alpha}^{i}$ may be
expressed as a $\mathbb{Z}$ linear combination^{} of lower powers of
$\alpha $. In other $\alpha $ solves a monic polynomial of
degree $i$ with coefficients in $\mathbb{Z}$.

Given a conjugacy class^{} $C$ in $G$, we may set ${\varphi}_{C}={\sum}_{g\in C}g$. Then ${\varphi}_{C}$ is central in $\mathbb{Z}G$, as given $h\in G$, we have:

$${\varphi}_{C}h=h\sum _{g\in C}{h}^{-1}gh=h\sum _{g\in C}g=h{\varphi}_{C}$$ |

Hence applying ${\varphi}_{C}$ to $V$ induces a $\u2102G$ linear map
$V\to V$. By Schur’s lemma this must be
multiplication by some
complex number^{} ${\lambda}_{C}$. Then ${\lambda}_{C}$ is an algebraic
integer^{} as it solves the same monic polynomial as ${\varphi}_{C}$.

Also any $g\in G$ has finite order so the map it induces on $V$
must have eigenvalues^{} which are roots of unity^{} and hence algebraic
integers. Hence the sum of the eigenvalues, ${\chi}_{V}(g)$, must
also be an algebraic integer.

Now $V$ is irreducible so,

$$|G|=\sum _{g\in G}{\chi}_{V}(g){\chi}_{V}{(g)}^{*}=\sum _{C\subset G}\mathrm{tr}({\varphi}_{C}){\chi}_{V}{(C)}^{*}=d\sum _{C\subset G}{\lambda}_{C}{\chi}_{V}{(C)}^{*}$$ |

Therefore $|G|/d$ is both rational and an algebraic integer. Hence it is an integer and $d$ divides $|G|$.

Title | proof that dimension of complex irreducible representation divides order of group |
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Canonical name | ProofThatDimensionOfComplexIrreducibleRepresentationDividesOrderOfGroup |

Date of creation | 2013-03-22 17:09:04 |

Last modified on | 2013-03-22 17:09:04 |

Owner | whm22 (2009) |

Last modified by | whm22 (2009) |

Numerical id | 8 |

Author | whm22 (2009) |

Entry type | Proof |

Classification | msc 20C99 |