Let $\chi$ be a nontrivial irreducible character of finite group $G$, and $G$ has odd order. Then, $\chi$ isn't equal to $\bar \chi$.
2026-03-25 22:01:52.1774476112
Character theory question.
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Hint: using the orthogonality relations, show that if $\chi \neq 1_G$ (the trivial character) and $\chi=\bar{\chi}$, then $\chi(1)=2\alpha$ for some algebraic integer $\alpha$.