# Tagged Questions

For questions about characters (homomorphisms from a group into the multiplicative group of a field).

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### Why does complex conjugation permute the rows (columns) of a character table

If $\chi$ is the character of $\rho$, then $\overline{\chi}$ is the character of $\rho^*$ (dual) and $\chi_{irreducible} \iff \overline{\chi_{irreducible}}$. This implies complex conjugation ...
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### Find the character of $\mathbb{C}[S_4/D_8]$

Find the character of $\mathbb{C}[S_4/D_8]$. I am assuming with this question that the first step will be to compute the (left) cosets $\{ gD_8: g \in S_4 \}$. Then I'm assuming that then it will ...
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### Understanding restriction in $S_3$

Let $G=S_3$. \begin{array}{|c|c|c|} \hline & e & (123) & (12) \\ \hline \chi_0 & 1 &1 & 1\\ \hline \chi_1 & 1 & 1 & -1 \\ \hline \chi_2 & 2 & -1 & 0 \\ ...
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### How much we know about the Group from its Complex character table?

Suppose $G$ is a finite group and suppose that complex character table of $G$ is given.It is well known that from character table we cannot determine the Group uniquely (For example $Q_8$ and $D_8$ ...
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### Character theory - Exercise 5.14

I am trying to solve the exercise 5.14 from the Isaac Martins Character Theory of Finite Groups. Let $G$ be a nonabelian group and let $f=min\{\chi(1) | \chi \in Irr(G), \chi(1)>1 \}.$ Show ...
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### Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ \chi(g)$

Suppose $\chi$ is an irreducible character of $G$. Suppose $z ∈ Z(G)$ and that $z$ has order $m$. Prove that there exists an $m$th root of unity $λ ∈ C$ such that for all $g ∈ G$, $\chi(zg) =λ \chi(g)$...