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1. (10 marks) Let $\chi$ be a complex character of $G$. The goal of this exercise is to show that for all $g \in G$, $\chi(g^{-1}) = \chi(g)$. (a) Let $\rho: G \to GL(V)$ be the representation of $G$ corresponding to $\chi$. Let $g \in G$ and set $k = |g|$ (the order of $g$). Show that the minimal polynomial of $\rho(g)$ divides $x^k - 1$. (b) What does (a) tell us about the eigenvalues of $\rho(g)$? (c) Use (a) and (b) to show that $\chi(g)$ is a sum of roots of unity in $\mathbb{C}$. (d) Let $\zeta$ be a $k$-th root of unity in $\mathbb{C}$. What is $\zeta^{-1}$? (e) Use (c) and (d) to show that $\chi(g^{-1}) = \chi(g)$.

          1. (10 marks) Let $\chi$ be a complex character of $G$. The goal of this exercise is to show that for all $g \in G$, $\chi(g^{-1}) = \chi(g)$.
(a) Let $\rho: G \to GL(V)$ be the representation of $G$ corresponding to $\chi$. Let $g \in G$ and set $k = |g|$ (the order of $g$). Show that the minimal polynomial of $\rho(g)$ divides $x^k - 1$.
(b) What does (a) tell us about the eigenvalues of $\rho(g)$?
(c) Use (a) and (b) to show that $\chi(g)$ is a sum of roots of unity in $\mathbb{C}$.
(d) Let $\zeta$ be a $k$-th root of unity in $\mathbb{C}$. What is $\zeta^{-1}$?
(e) Use (c) and (d) to show that $\chi(g^{-1}) = \chi(g)$.

1. (10 marks) Let χ be a complex character of G. The goal of this exercise is to show that for all g ∈ G, χ(g^-1) = χ(g).
(a) Let ρ: G → GL(V) be the representation of G corresponding to χ. Let g ∈ G and set k = |g| (the order of g). Show that the minimal polynomial of ρ(g) divides x^k - 1.
(b) What does (a) tell us about the eigenvalues of ρ(g)?
(c) Use (a) and (b) to show that χ(g) is a sum of roots of unity in ℂ.
(d) Let ζ be a k-th root of unity in ℂ. What is ζ^-1?
(e) Use (c) and (d) to show that χ(g^-1) = χ(g).

Added by Debbie D.

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