Let $G$ be a finite group and $C$ the center of $G$. Let$μ:C→F^×$ be a character of $C$. Prove that there is an irreducible representation $ρ : G → GL(V )$ such that $ρ(c)(v) = μ(c)v$ for all $c ∈ C$ and $v ∈ V$ .

If the field is algebraically closed, then the representation space must be $1-dimensional$. So making elements in $G-C$ act trivially gives an irreducible representation. But for an arbitrary field, I don't even know whether I can apply $<\chi_\rho,\chi_\rho>=1$ iff $\rho$ is irreducible,because it seems that the theorem is based on algebraic closedness of the base field. I know that some other construction can make an extension, like induced representations in wikipedia. But again, I am not sure whether the extension is irreducible.

  • $\begingroup$ Try induction from the center. $\endgroup$ – Qiaochu Yuan May 12 '16 at 6:36
  • $\begingroup$ How to check whether I get an irreducible representation, since I may not be able to use orthogonal relation? $\endgroup$ – user280486 May 12 '16 at 15:45
  • $\begingroup$ I think I can extend the representation first, regardless of irreducibility of the representation space. Since elements in the center still act as scalar multiplications on an irreducible component, I may take an irreducible component and get an irreducible representation. $\endgroup$ – user280486 May 12 '16 at 15:54
  • $\begingroup$ And such an extension can be done by using induced representation. Does my solution work? $\endgroup$ – user280486 May 12 '16 at 17:23

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