Standing hypotheses for ordinary character theory: $G$ finite, $k=\mathbb{C}$, and every representation finite-dimensional

The ordinary-character-theory items on this page work inside the following setting, fixed once here: $G$ is a finite group, the base field is $k=\mathbb{C}$, and every representation is finite-dimensional (A finite-dimensional representation $\rho :G\to \mathrm{GL}(V)$ over a field, and its degree). This is ordinary character theory in the sense of Webb, Chapter 3 and Etingof et al., Section 3.3: no infinite group, unitary-representation, or modular-character material is load-bearing in the character-theoretic arguments on the page. The quotient-factorisation result A representation with kernel containing a normal subgroup factors through the quotient, and irreducibility is unchanged by inflation is deliberately stated in the greater generality of an arbitrary group, field, and representation because its proof needs none of these standing restrictions.

The choice of $k=\mathbb{C}$ is what makes the hypotheses of the published representation-theory spine available. Since $\mathrm{char}\mathbb{C}=0$ and $G$ is finite, the characteristic does not divide $\mathrm{\mid }G\mathrm{\mid }$, so every finite-dimensional representation is completely reducible (If $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, every finite-dimensional representation of $G$ is completely reducible), and the field is algebraically closed, which feeds the count of irreducibles against conjugacy classes (If $k$ is algebraically closed and $\mathrm{char}k\nmid \mathrm{\mid }G\mathrm{\mid }$, the number of irreducible representations of $G$ equals the number of conjugacy classes). These hypotheses are restated where they are consumed; the present remark fixes the default scope without narrowing an item that explicitly states broader hypotheses.