每日定理11

Isaacs, $\textit{Character Theory of Finite Groups}$, Lemma(2.3)

  1. Similar $F$-representations of $G$ afford equal characters.
  2. Characters are constant on conjugacy classes of a group.

Pf:

  • $tr(P^{-1}AP)=tr(A)$
  • $\mathfrak{X}(h^{-1}gh)=\mathfrak{X}(h)^{-1}\mathfrak{X}(g)\mathfrak{X}(h)$, and hence $tr(\mathfrak{X}(h^{-1}gh))=tr(\mathfrak{X}(g))$
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