Isaacs, $\textit{Character Theory of Finite Groups}$, Corollary(2.9)

Let $\mathfrak{X}$ and $\mathfrak{Y}$ be $\mathbb{C}$-representations of a group. Then $\mathfrak{X}$ and $\mathfrak{Y}$ are similar iff they afford equal characters.

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Pf:

  • $(\Rightarrow)$ Obviously by Lemma(2.3).

  •  $(\Leftarrow)$ $\mathfrak{X}$ and $\mathfrak{Y}$ correspond to $V$ and $W$, respectively. Then $\mathfrak{X}$ affords $\sum n_{M_i}(V)\chi_i$ and $\mathfrak{Y}$ affords $\sum n_{M_i}(W)\chi_i$.

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