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Let $D_5$ be the dihedral group on $5$ vertices, and let $C_5$ a cyclic group of order $5$.

I have determined the character tables for both $D_5$ and $C_5$, and am now trying to decompose the restriction of each irreducible character of $D_5$ into irreducible characters of $C_5.$

My problem is that I'm not sure what is meant by this decomposition. Could anyone illuminate this problem?

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Hint : I'll take $C_5 = \langle r \rangle \subset D_5$.

Any representation of $D_5$ restricts to a representation of $C_5$. This representation of $C_5$ will not stay irreducible anymore e.g if the dimension of your representation is $2$. If $V$ was your representation of $D_5$, then $\chi_{V|C_5}(r^k) = \chi_V(r^k)$ by definition. Since you already know the character table for $C_5$, you can now easily split $\chi_{V | C_5}$ into irreducible characters.

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  • $\begingroup$ What happens to the one dimensional representations of $\chi_V$? $\endgroup$ – user352541 Apr 23 '17 at 14:33
  • $\begingroup$ What does mean "representation of $\chi_V$" ? $V$ is a fixed representation. $\endgroup$ – user171326 Apr 23 '17 at 14:48

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