Remark 4

Tag 00O7, permanent. Cite it as [Bourbaki, Tag 00O7].

Algebra, VIII, p. 420, § 21. Linear Representations of Finite Groups

Let $H$ be a subgroup of $G$ of cardinal $m$ . The isomorphism $\varphi$ restricts to an isomorphism from $\mu_m(K)$ to $\mu_m(K')$ and, consequently, a ring isomorphism $\varphi_H$ from $R_K(H)$ to $R_{K'}(H)$ . The following diagrams commute:

\begin{array}{ccc} R_K(G) & \xrightarrow{\operatorname{Res}_H^G} & R_K(H) \\ \downarrow \varphi_G & & \downarrow \varphi_H \\ R_{K'}(G) & \xrightarrow{\operatorname{Res}_H^{G'}} & R_{K'}(H) , \end{array}

\begin{array}{ccc} R_K(H) & \xrightarrow{\operatorname{Ind}_H^G} & R_K(G) \\ \downarrow \varphi_H & & \downarrow \varphi_G \\ R_{K'}(H) & \xrightarrow{\operatorname{Ind}_H^{G'}} & R_{K'}(G) . \end{array}

The commutativity of the first diagram is obvious, and that of the second follows from it using Frobenius reciprocity and formula (47).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.